{"id":3303,"date":"2026-01-28T18:00:42","date_gmt":"2026-01-28T12:30:42","guid":{"rendered":"https:\/\/www.manabadi.co.in\/boards\/?p=3303"},"modified":"2026-01-30T14:36:58","modified_gmt":"2026-01-30T09:06:58","slug":"ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6c","status":"publish","type":"post","link":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6c\/","title":{"rendered":"TS Inter 1st Year Maths 1A Trigonometric Ratios upto Transformations Solutions Exercise 6(c)"},"content":{"rendered":"\n<h3>I.<br> Question 1.<br> Simplify the following<br> (i) cos 100\u00b0 . cos 40\u00b0 + sin 100\u00b0 . sin 40\u00b0<\/h3>\n\n<p>Answer:<br> Use cos A. cos B + sin A sin B = cos (A \u2013 B)<br> \u2234 cos 100\u00b0 . cos 40\u00b0 + sin 100\u00b0.sin 40\u00b0<br> = cos (100\u00b0 \u2013 40\u00b0)<br> = cos 60\u00b0<br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-1-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2\"><span class=\"mfrac\" id=\"MathJax-Span-3\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-4\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-5\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-1\">\\frac{1}{2}<\/script> = R.H.S<\/p>\n\n<p>(ii) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-2-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-6\" style=\"width: 6.666em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.734em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1005.73em, 2.891em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-7\"><span class=\"mfrac\" id=\"MathJax-Span-8\"><span style=\"display: inline-block; position: relative; width: 5.454em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1005.32em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -2.654em;\"><span class=\"mrow\" id=\"MathJax-Span-9\"><span class=\"mi\" id=\"MathJax-Span-10\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cot<\/span><span class=\"mo\" id=\"MathJax-Span-11\" style=\"font-size: 70.7%;\"><\/span><span class=\"msubsup\" id=\"MathJax-Span-12\" style=\"padding-left: 0.189em;\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.65em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-13\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">55<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.285em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-14\"><span class=\"mrow\" id=\"MathJax-Span-15\"><span class=\"mo\" id=\"MathJax-Span-16\" style=\"font-size: 50%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-17\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">cot<\/span><span class=\"mo\" id=\"MathJax-Span-18\" style=\"font-size: 70.7%;\"><\/span><span class=\"msubsup\" id=\"MathJax-Span-19\" style=\"padding-left: 0.189em;\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.65em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-20\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">35<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.285em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-21\"><span class=\"mrow\" id=\"MathJax-Span-22\"><span class=\"mo\" id=\"MathJax-Span-23\" style=\"font-size: 50%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-24\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-25\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1004.85em, 4.196em, -999.998em); top: -3.586em; left: 50%; margin-left: -2.421em;\"><span class=\"mrow\" id=\"MathJax-Span-26\"><span class=\"mi\" id=\"MathJax-Span-27\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cot<\/span><span class=\"mo\" id=\"MathJax-Span-28\" style=\"font-size: 70.7%;\"><\/span><span class=\"msubsup\" id=\"MathJax-Span-29\" style=\"padding-left: 0.189em;\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.65em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-30\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">55<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.285em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-31\"><span class=\"mrow\" id=\"MathJax-Span-32\"><span class=\"mo\" id=\"MathJax-Span-33\" style=\"font-size: 50%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-34\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-35\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cot<\/span><span class=\"mo\" id=\"MathJax-Span-36\" style=\"font-size: 70.7%;\"><\/span><span class=\"msubsup\" id=\"MathJax-Span-37\" style=\"padding-left: 0.189em;\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.65em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-38\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">35<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.285em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-39\"><span class=\"mrow\" id=\"MathJax-Span-40\"><span class=\"mo\" id=\"MathJax-Span-41\" style=\"font-size: 50%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1005.45em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 5.454em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.592em; border-left: 0px solid; width: 0px; height: 1.732em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-2\">\\frac{\\cot 55^{\\circ} \\cot 35^{\\circ}-1}{\\cot 55^{\\circ}+\\cot 35^{\\circ}}<\/script><br>\n\n<p>Answer:<br> <img decoding=\"async\" class=\"alignnone size-full wp-image-7664\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS_Inter_1st_Year_Maths_1A_Solutions_Chapter_6_Trigonometric_Ratios.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 1\" width=\"284\" height=\"120\" data-pin-nopin=\"true\"><\/p>\n\n<p>(iii) tan [<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-3-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-42\" style=\"width: 0.934em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.795em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.447em, 1000.79em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-43\"><span class=\"mfrac\" id=\"MathJax-Span-44\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.42em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-45\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-46\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.51em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.515em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-3\">\\frac{\\pi}{4}<\/script> + \u03b8]. tan[<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-4-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-47\" style=\"width: 0.934em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.795em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.447em, 1000.79em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-48\"><span class=\"mfrac\" id=\"MathJax-Span-49\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.42em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-50\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-51\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.51em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.515em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-4\">\\frac{\\pi}{4}<\/script> \u2013 \u03b8]<br>\n\nAnswer:<br> <img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-7663\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6c-2.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 2\" width=\"321\" height=\"257\" sizes=\"(max-width: 321px) 100vw, 321px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 2\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c)\"><\/p>\n\n\n\n<p>(iv) Evaluate \u03a3<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-5-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-52\" style=\"width: 6.06em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.221em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1005.22em, 2.845em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-53\"><span class=\"mfrac\" id=\"MathJax-Span-54\"><span style=\"display: inline-block; position: relative; width: 4.942em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.264em, 1004.8em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -2.421em;\"><span class=\"mrow\" id=\"MathJax-Span-55\"><span class=\"msubsup\" id=\"MathJax-Span-56\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.93em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-57\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.285em; left: 0.934em;\"><span class=\"mn\" id=\"MathJax-Span-58\" style=\"font-size: 50%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-59\" style=\"font-size: 70.7%;\"><\/span><span class=\"texatom\" id=\"MathJax-Span-60\" style=\"padding-left: 0.189em;\"><span class=\"mrow\" id=\"MathJax-Span-61\"><span class=\"mtext\" id=\"MathJax-Span-62\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">&nbsp;<\/span><span class=\"mi\" id=\"MathJax-Span-63\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">B<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-64\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"msubsup\" id=\"MathJax-Span-65\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.93em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-66\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.285em; left: 0.934em;\"><span class=\"mn\" id=\"MathJax-Span-67\" style=\"font-size: 50%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-68\" style=\"font-size: 70.7%;\"><\/span><span class=\"texatom\" id=\"MathJax-Span-69\" style=\"padding-left: 0.189em;\"><span class=\"mrow\" id=\"MathJax-Span-70\"><span class=\"mtext\" id=\"MathJax-Span-71\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">&nbsp;<\/span><span class=\"mi\" id=\"MathJax-Span-72\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">A<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1004.43em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -2.234em;\"><span class=\"mrow\" id=\"MathJax-Span-73\"><span class=\"msubsup\" id=\"MathJax-Span-74\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.93em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-75\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.192em; left: 0.934em;\"><span class=\"mn\" id=\"MathJax-Span-76\" style=\"font-size: 50%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-77\" style=\"font-size: 70.7%;\"><\/span><span class=\"texatom\" id=\"MathJax-Span-78\" style=\"padding-left: 0.189em;\"><span class=\"mrow\" id=\"MathJax-Span-79\"><span class=\"mtext\" id=\"MathJax-Span-80\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">&nbsp;<\/span><span class=\"mi\" id=\"MathJax-Span-81\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">A<\/span><\/span><\/span><span class=\"msubsup\" id=\"MathJax-Span-82\" style=\"padding-left: 0.189em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.93em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-83\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.192em; left: 0.934em;\"><span class=\"mn\" id=\"MathJax-Span-84\" style=\"font-size: 50%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-85\" style=\"font-size: 70.7%;\"><\/span><span class=\"texatom\" id=\"MathJax-Span-86\" style=\"padding-left: 0.189em;\"><span class=\"mrow\" id=\"MathJax-Span-87\"><span class=\"mtext\" id=\"MathJax-Span-88\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">&nbsp;<\/span><span class=\"mi\" id=\"MathJax-Span-89\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">B<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1004.94em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.942em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.538em; border-left: 0px solid; width: 0px; height: 1.841em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-5\">\\frac{\\cos ^2 \\mathrm{~B}-\\cos ^2 \\mathrm{~A}}{\\cos ^2 \\mathrm{~A} \\cos ^2 \\mathrm{~B}}<\/script> if none of sin A, sin B, sin C is zero.<br> Answer:<br> \u03a3<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-6-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-90\" style=\"width: 6.06em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.221em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1005.22em, 2.845em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-91\"><span class=\"mfrac\" id=\"MathJax-Span-92\"><span style=\"display: inline-block; position: relative; width: 4.942em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.264em, 1004.8em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -2.421em;\"><span class=\"mrow\" id=\"MathJax-Span-93\"><span class=\"msubsup\" id=\"MathJax-Span-94\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.93em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-95\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.285em; left: 0.934em;\"><span class=\"mn\" id=\"MathJax-Span-96\" style=\"font-size: 50%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-97\" style=\"font-size: 70.7%;\"><\/span><span class=\"texatom\" id=\"MathJax-Span-98\" style=\"padding-left: 0.189em;\"><span class=\"mrow\" id=\"MathJax-Span-99\"><span class=\"mtext\" id=\"MathJax-Span-100\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">&nbsp;<\/span><span class=\"mi\" id=\"MathJax-Span-101\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">B<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-102\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"msubsup\" id=\"MathJax-Span-103\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.93em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-104\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.285em; left: 0.934em;\"><span class=\"mn\" id=\"MathJax-Span-105\" style=\"font-size: 50%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-106\" style=\"font-size: 70.7%;\"><\/span><span class=\"texatom\" id=\"MathJax-Span-107\" style=\"padding-left: 0.189em;\"><span class=\"mrow\" id=\"MathJax-Span-108\"><span class=\"mtext\" id=\"MathJax-Span-109\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">&nbsp;<\/span><span class=\"mi\" id=\"MathJax-Span-110\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">A<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1004.43em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -2.234em;\"><span class=\"mrow\" id=\"MathJax-Span-111\"><span class=\"msubsup\" id=\"MathJax-Span-112\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.93em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-113\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.192em; left: 0.934em;\"><span class=\"mn\" id=\"MathJax-Span-114\" style=\"font-size: 50%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-115\" style=\"font-size: 70.7%;\"><\/span><span class=\"texatom\" id=\"MathJax-Span-116\" style=\"padding-left: 0.189em;\"><span class=\"mrow\" id=\"MathJax-Span-117\"><span class=\"mtext\" id=\"MathJax-Span-118\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">&nbsp;<\/span><span class=\"mi\" id=\"MathJax-Span-119\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">A<\/span><\/span><\/span><span class=\"msubsup\" id=\"MathJax-Span-120\" style=\"padding-left: 0.189em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.93em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-121\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.192em; left: 0.934em;\"><span class=\"mn\" id=\"MathJax-Span-122\" style=\"font-size: 50%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-123\" style=\"font-size: 70.7%;\"><\/span><span class=\"texatom\" id=\"MathJax-Span-124\" style=\"padding-left: 0.189em;\"><span class=\"mrow\" id=\"MathJax-Span-125\"><span class=\"mtext\" id=\"MathJax-Span-126\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">&nbsp;<\/span><span class=\"mi\" id=\"MathJax-Span-127\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">B<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1004.94em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.942em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.538em; border-left: 0px solid; width: 0px; height: 1.841em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-6\">\\frac{\\cos ^2 \\mathrm{~B}-\\cos ^2 \\mathrm{~A}}{\\cos ^2 \\mathrm{~A} \\cos ^2 \\mathrm{~B}}<\/script> = \u03a3<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-7-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-128\" style=\"width: 10.347em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 8.903em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.866em, 1008.72em, 3.963em, -999.998em); top: -3.166em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-129\"><span class=\"mrow\" id=\"MathJax-Span-130\"><span class=\"mo\" id=\"MathJax-Span-131\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-132\"><span style=\"display: inline-block; position: relative; width: 7.411em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1007.32em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -3.632em;\"><span class=\"mrow\" id=\"MathJax-Span-133\"><span class=\"mi\" id=\"MathJax-Span-134\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-135\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-136\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">C<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-137\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-138\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-139\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-140\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-141\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-142\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-143\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">C<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-144\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-145\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-146\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">A<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1003.31em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -1.629em;\"><span class=\"mrow\" id=\"MathJax-Span-147\"><span class=\"mi\" id=\"MathJax-Span-148\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-149\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-150\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">C<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-151\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-152\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-153\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">A<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1007.41em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 7.411em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-154\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.171em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.808em; border-left: 0px solid; width: 0px; height: 2.219em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-7\">\\left(\\frac{\\sin C \\cos A-\\cos C \\sin A}{\\sin C \\sin A}\\right)<\/script><br> = \u03a3 (cot A \u2013 cot C)<br> = cot A \u2013 cot C + cot B \u2013 cot A + cot C \u2013 cot B = 0<\/p>\n\n<h3>Question 4.<br> (i) Prove that<br> cos 35\u00b0 + cos 85\u00b0 + cos 155\u00b0 = 0<\/h3>\n\n<p>Answer:<br> cos 35\u00b0 + cos 85\u00b0 + cos 155\u00b0<br> = cos 35\u00b0 + 2cos<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-8-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-155\" style=\"width: 4.569em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.917em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.866em, 1003.73em, 3.963em, -999.998em); top: -3.166em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-156\"><span class=\"mrow\" id=\"MathJax-Span-157\"><span class=\"mo\" id=\"MathJax-Span-158\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-159\"><span style=\"display: inline-block; position: relative; width: 2.425em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1002.29em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -1.163em;\"><span class=\"mrow\" id=\"MathJax-Span-160\"><span class=\"mn\" id=\"MathJax-Span-161\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">85<\/span><span class=\"mo\" id=\"MathJax-Span-162\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-163\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">155<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-164\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1002.42em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.425em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-165\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.171em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.808em; border-left: 0px solid; width: 0px; height: 2.219em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-8\">\\left(\\frac{85+155}{2}\\right)<\/script>cos<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-9-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-166\" style=\"width: 4.569em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.917em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.866em, 1003.73em, 3.963em, -999.998em); top: -3.166em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-167\"><span class=\"mrow\" id=\"MathJax-Span-168\"><span class=\"mo\" id=\"MathJax-Span-169\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-170\"><span style=\"display: inline-block; position: relative; width: 2.425em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1002.29em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -1.163em;\"><span class=\"mrow\" id=\"MathJax-Span-171\"><span class=\"mn\" id=\"MathJax-Span-172\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">85<\/span><span class=\"mo\" id=\"MathJax-Span-173\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-174\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">155<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-175\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1002.42em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.425em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-176\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.171em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.808em; border-left: 0px solid; width: 0px; height: 2.219em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-9\">\\left(\\frac{85-155}{2}\\right)<\/script><br> = cos 35\u00b0 + 2 cos 120\u00b0 cos (-35\u00b0)<br> = cos 35\u00b0 \u2013 cos 35\u00b0 = 0<\/p>\n\n<p>(ii) tan 72\u00b0 = tan 18\u00b0 + 2 tan 54\u00b0<br> Answer:<br> We have cot A \u2013 tan A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-10-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-177\" style=\"width: 2.425em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.099em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1002.1em, 2.845em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-178\"><span class=\"mfrac\" id=\"MathJax-Span-179\"><span style=\"display: inline-block; position: relative; width: 1.82em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-180\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.73em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.836em;\"><span class=\"mrow\" id=\"MathJax-Span-181\"><span class=\"mi\" id=\"MathJax-Span-182\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">tan<\/span><span class=\"mo\" id=\"MathJax-Span-183\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-184\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">A<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.82em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.82em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.538em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-10\">\\frac{1}{\\tan A}<\/script> \u2013 tan A<br> \u21d2 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-11-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-185\" style=\"width: 10.44em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 8.996em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(0.934em, 1009em, 2.891em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-186\"><span class=\"mfrac\" id=\"MathJax-Span-187\"><span style=\"display: inline-block; position: relative; width: 3.031em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.264em, 1002.89em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -1.442em;\"><span class=\"mrow\" id=\"MathJax-Span-188\"><span class=\"mn\" id=\"MathJax-Span-189\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-190\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"msubsup\" id=\"MathJax-Span-191\"><span style=\"display: inline-block; position: relative; width: 1.307em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1001.03em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-192\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">tan<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.285em; left: 1.027em;\"><span class=\"mn\" id=\"MathJax-Span-193\" style=\"font-size: 50%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-194\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-195\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">A<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.73em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.836em;\"><span class=\"mrow\" id=\"MathJax-Span-196\"><span class=\"mi\" id=\"MathJax-Span-197\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">tan<\/span><span class=\"mo\" id=\"MathJax-Span-198\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-199\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">A<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1003.03em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.031em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-200\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-201\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 4.056em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.264em, 1003.82em, 4.383em, -999.998em); top: -4.658em; left: 50%; margin-left: -1.955em;\"><span class=\"mrow\" id=\"MathJax-Span-202\"><span class=\"mn\" id=\"MathJax-Span-203\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mrow\" id=\"MathJax-Span-204\"><span class=\"mo\" id=\"MathJax-Span-205\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size1;\">(<\/span><\/span><\/span><span class=\"mn\" id=\"MathJax-Span-206\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-207\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"msubsup\" id=\"MathJax-Span-208\"><span style=\"display: inline-block; position: relative; width: 1.307em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1001.03em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-209\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">tan<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.285em; left: 1.027em;\"><span class=\"mn\" id=\"MathJax-Span-210\" style=\"font-size: 50%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-211\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-212\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-213\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size1;\">)<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1002.24em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -1.116em;\"><span class=\"mrow\" id=\"MathJax-Span-214\"><span class=\"mn\" id=\"MathJax-Span-215\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-216\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">tan<\/span><span class=\"mo\" id=\"MathJax-Span-217\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-218\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">A<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1004.06em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.056em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.538em; border-left: 0px solid; width: 0px; height: 2.057em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-11\">\\frac{1-\\tan ^2 A}{\\tan A}=\\frac{2\\left(1-\\tan ^2 A\\right)}{2 \\tan A}<\/script> = 2 cot 2A<br> \u2234 cot A \u2013 tan A = 2 cot 2A<br> \u21d2 cot A = tan A + 2 cot 2A\u00b0<br> Take A = 18\u00b0, then cot 18\u00b0 = tan 18\u00b0 + 2 cot 36\u00b0<br> \u21d2 cot(90 \u2013 72) = tan 18\u00b0 + 2 cot(90 \u2013 54)<br> \u21d2 tan 72\u00b0 = tan 18\u00b0 + 2 tan 54\u00b0<\/p>\n\n\n\n<p>(iii) sin 750\u00b0 cos 480\u00b0 + cos 120\u00b0 cos 60\u00b0 = \u2013<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-12-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-219\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-220\"><span class=\"mfrac\" id=\"MathJax-Span-221\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-222\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-223\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-12\">\\frac{1}{2}<\/script><br> Answer:<br> L.H.S = sin [2.(360) + 30] cos[360 + 120] + cos 120 cos 60<br> = sin 30 cos 120 + cos 120 cos 60<br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-13-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-224\" style=\"width: 14.774em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 12.724em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.4em, 1012.72em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-225\"><span class=\"mrow\" id=\"MathJax-Span-226\"><span class=\"mo\" id=\"MathJax-Span-227\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-228\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-229\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-230\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-231\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">)<\/span><\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-232\" style=\"padding-left: 0.189em;\"><span class=\"mo\" id=\"MathJax-Span-233\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mo\" id=\"MathJax-Span-234\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-235\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-236\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-237\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-238\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">)<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-239\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mrow\" id=\"MathJax-Span-240\" style=\"padding-left: 0.235em;\"><span class=\"mo\" id=\"MathJax-Span-241\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mo\" id=\"MathJax-Span-242\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-243\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-244\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-245\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-246\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">)<\/span><\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-247\" style=\"padding-left: 0.189em;\"><span class=\"mo\" id=\"MathJax-Span-248\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-249\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-250\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-251\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-252\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">)<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-253\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mo\" id=\"MathJax-Span-254\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-255\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-256\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-257\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.425em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-13\">\\left(\\frac{1}{2}\\right)\\left(-\\frac{1}{2}\\right)+\\left(-\\frac{1}{2}\\right)\\left(\\frac{1}{2}\\right)=-\\frac{1}{2}<\/script><\/p>\n\n<p>(iv) cos A + cos(<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-14-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-258\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-259\"><span class=\"mfrac\" id=\"MathJax-Span-260\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.75em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-261\"><span class=\"mn\" id=\"MathJax-Span-262\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span class=\"mi\" id=\"MathJax-Span-263\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-264\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-14\">\\frac{4 \\pi}{3}<\/script> \u2013 A) + cos(<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-15-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-265\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-266\"><span class=\"mfrac\" id=\"MathJax-Span-267\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.75em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-268\"><span class=\"mn\" id=\"MathJax-Span-269\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span class=\"mi\" id=\"MathJax-Span-270\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-271\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-15\">\\frac{4 \\pi}{3}<\/script> + A) = 0<br> Answer:<br> Use cos(A + B) + cos(A \u2013 B) = 2cos A cos B<br> L.H.S = cos A + 2cos<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-16-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-272\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-273\"><span class=\"mfrac\" id=\"MathJax-Span-274\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.75em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-275\"><span class=\"mn\" id=\"MathJax-Span-276\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span class=\"mi\" id=\"MathJax-Span-277\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-278\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-16\">\\frac{4 \\pi}{3}<\/script>cos A<br> = cos A + 2 cos 240 cos A<br> = cos A + 2 cos ( 180 + 60) cos A<br> = cos A + 2 ( \u2013 cos 60) cos A<br> = cos A + 2<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-17-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-279\" style=\"width: 2.891em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.472em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.4em, 1002.33em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-280\"><span class=\"mrow\" id=\"MathJax-Span-281\"><span class=\"mo\" id=\"MathJax-Span-282\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mo\" id=\"MathJax-Span-283\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-284\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-285\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-286\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-287\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">)<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.425em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-17\">\\left(-\\frac{1}{2}\\right)<\/script>cos A = cos A \u2013 cos A = 0<\/p>\n\n<p>(v) cos<sup>2<\/sup>\u03b8 + cos<sup>2<\/sup>(<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-18-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-288\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-289\"><span class=\"mfrac\" id=\"MathJax-Span-290\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.75em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-291\"><span class=\"mn\" id=\"MathJax-Span-292\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-293\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-294\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-18\">\\frac{2 \\pi}{3}<\/script> + \u03b8) + cos<sup>2<\/sup>(<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-19-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-295\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-296\"><span class=\"mfrac\" id=\"MathJax-Span-297\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.75em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-298\"><span class=\"mn\" id=\"MathJax-Span-299\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-300\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-301\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-19\">\\frac{2 \\pi}{3}<\/script> \u2013 \u03b8) = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-20-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-302\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-303\"><span class=\"mfrac\" id=\"MathJax-Span-304\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-305\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-306\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-20\">\\frac{3}{2}<\/script><br> Answer:<br> cos<sup>2<\/sup>\u03b8 + cos<sup>2<\/sup>( 120 + \u03b8) + cos<sup>2<\/sup>( 120 \u2013 \u03b8)<br> = cos<sup>2<\/sup>\u03b8 + cos<sup>2<\/sup>(120 + \u03b8) + 1 \u2013 sin<sup>2<\/sup>(120 \u2013 \u03b8)<br> = 1 + cos<sup>2<\/sup>\u03b8 + cos [ 120 + \u03b8 + 120 \u2013 \u03b8] cos [ 120 + \u03b8- 120 + \u03b8]<br> = 1 + cos<sup>2<\/sup>\u03b8 + cos (240) cos 2\u03b8<br> = 1 + cos<sup>2<\/sup>\u03b8 + cos (180 + 60) cos 2\u03b8<br> [\u2235 Use cos<sup>2<\/sup>A \u2013 sin<sup>2<\/sup>B = cos (A + B) cos (A \u2013 B)]<br> = 1 + cos<sup>2<\/sup>\u03b8 \u2013 cos 60 ( 2 cos2\u03b8 \u2013 1)<br> = 1 + cos<sup>2<\/sup>\u03b8 \u2013 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-21-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-307\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-308\"><span class=\"mfrac\" id=\"MathJax-Span-309\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-310\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-311\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-21\">\\frac{1}{2}<\/script> ( 2 cos2\u03b8 \u2013 1)<br> = 1 + cos<sup>2<\/sup>\u03b8 \u2013 cos<sup>2<\/sup> \u03b8 + <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-22-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-312\" style=\"width: 3.311em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.845em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.307em, 1002.85em, 2.845em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-313\"><span class=\"mfrac\" id=\"MathJax-Span-314\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-315\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-316\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-317\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-318\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-319\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-320\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-22\">\\frac{1}{2}=\\frac{3}{2}<\/script><\/p>\n\n\n\n<h3>Question 5.<br> Evaluate<br> (i) sin<sup>2<\/sup>82<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-23-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-321\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-322\"><span class=\"msubsup\" id=\"MathJax-Span-323\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-324\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-325\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-326\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-327\"><span class=\"mrow\" id=\"MathJax-Span-328\"><span class=\"mo\" id=\"MathJax-Span-329\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-23\">\\frac{1}{2}^{\\circ}<\/script> \u2013 sin<sup>2<\/sup>22<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-24-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-330\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-331\"><span class=\"msubsup\" id=\"MathJax-Span-332\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-333\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-334\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-335\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-336\"><span class=\"mrow\" id=\"MathJax-Span-337\"><span class=\"mo\" id=\"MathJax-Span-338\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-24\">\\frac{1}{2}^{\\circ}<\/script><\/h3>\n\n<p>Answer:<br> sin<sup>2<\/sup>82<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-25-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-339\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-340\"><span class=\"msubsup\" id=\"MathJax-Span-341\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-342\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-343\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-344\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-345\"><span class=\"mrow\" id=\"MathJax-Span-346\"><span class=\"mo\" id=\"MathJax-Span-347\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-25\">\\frac{1}{2}^{\\circ}<\/script> \u2013 sin<sup>2<\/sup>22<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-26-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-348\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-349\"><span class=\"msubsup\" id=\"MathJax-Span-350\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-351\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-352\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-353\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-354\"><span class=\"mrow\" id=\"MathJax-Span-355\"><span class=\"mo\" id=\"MathJax-Span-356\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-26\">\\frac{1}{2}^{\\circ}<\/script><br> = sin[82<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-27-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-357\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-358\"><span class=\"msubsup\" id=\"MathJax-Span-359\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-360\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-361\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-362\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-363\"><span class=\"mrow\" id=\"MathJax-Span-364\"><span class=\"mo\" id=\"MathJax-Span-365\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-27\">\\frac{1}{2}^{\\circ}<\/script> + 22<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-28-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-366\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-367\"><span class=\"msubsup\" id=\"MathJax-Span-368\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-369\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-370\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-371\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-372\"><span class=\"mrow\" id=\"MathJax-Span-373\"><span class=\"mo\" id=\"MathJax-Span-374\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-28\">\\frac{1}{2}^{\\circ}<\/script>]<br> \u2235 Use sin<sup>2<\/sup>A \u2013 sin<sup>2<\/sup>B = sin(A + B) sin(A \u2013 B)<br> = sin 105\u00b0 . sin 60\u00b0<br> = sin 60\u00b0 sin (60\u00b0 + 45\u00b0)<br> = sin 60\u00b0 [sin 60\u00b0 cos 45\u00b0 + cos 60\u00b0 sin 45\u00b0]<br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-29-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-375\" style=\"width: 18.082em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 15.566em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.866em, 1015.57em, 3.963em, -999.998em); top: -3.166em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-376\"><span class=\"mfrac\" id=\"MathJax-Span-377\"><span style=\"display: inline-block; position: relative; width: 1.074em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.311em, 1000.93em, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.464em;\"><span class=\"msqrt\" id=\"MathJax-Span-378\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-379\"><span class=\"mn\" id=\"MathJax-Span-380\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-381\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.07em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.074em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-382\"><span class=\"mo\" id=\"MathJax-Span-383\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">[<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-384\"><span style=\"display: inline-block; position: relative; width: 1.4em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.311em, 1000.93em, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.464em;\"><span class=\"msqrt\" id=\"MathJax-Span-385\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-386\"><span class=\"mn\" id=\"MathJax-Span-387\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1001.31em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -0.65em;\"><span class=\"mrow\" id=\"MathJax-Span-388\"><span class=\"mn\" id=\"MathJax-Span-389\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"msqrt\" id=\"MathJax-Span-390\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-391\"><span class=\"mn\" id=\"MathJax-Span-392\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.4em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.4em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-393\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mfrac\" id=\"MathJax-Span-394\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 1.4em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-395\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1001.31em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -0.65em;\"><span class=\"mrow\" id=\"MathJax-Span-396\"><span class=\"mn\" id=\"MathJax-Span-397\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"msqrt\" id=\"MathJax-Span-398\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-399\"><span class=\"mn\" id=\"MathJax-Span-400\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.4em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.4em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-401\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">]<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-402\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-403\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.311em, 1003.26em, 4.336em, -999.998em); top: -4.564em; left: 50%; margin-left: -1.675em;\"><span class=\"mrow\" id=\"MathJax-Span-404\"><span class=\"msqrt\" id=\"MathJax-Span-405\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-406\"><span class=\"mn\" id=\"MathJax-Span-407\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-408\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">(<\/span><span class=\"msqrt\" id=\"MathJax-Span-409\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-410\"><span class=\"mn\" id=\"MathJax-Span-411\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-412\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-413\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-414\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">)<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1001.31em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -0.65em;\"><span class=\"mrow\" id=\"MathJax-Span-415\"><span class=\"mn\" id=\"MathJax-Span-416\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span class=\"msqrt\" id=\"MathJax-Span-417\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-418\"><span class=\"mn\" id=\"MathJax-Span-419\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1003.45em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.451em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-420\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-421\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.959em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.311em, 1001.87em, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.93em;\"><span class=\"mrow\" id=\"MathJax-Span-422\"><span class=\"mn\" id=\"MathJax-Span-423\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"mo\" id=\"MathJax-Span-424\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"msqrt\" id=\"MathJax-Span-425\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-426\"><span class=\"mn\" id=\"MathJax-Span-427\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1001.31em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -0.65em;\"><span class=\"mrow\" id=\"MathJax-Span-428\"><span class=\"mn\" id=\"MathJax-Span-429\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span class=\"msqrt\" id=\"MathJax-Span-430\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-431\"><span class=\"mn\" id=\"MathJax-Span-432\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.96em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.959em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.171em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.808em; border-left: 0px solid; width: 0px; height: 2.219em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-29\">\\frac{\\sqrt{3}}{2}\\left[\\frac{\\sqrt{3}}{2 \\sqrt{2}}+\\frac{1}{2 \\sqrt{2}}\\right]=\\frac{\\sqrt{3}(\\sqrt{3}+1)}{4 \\sqrt{2}}=\\frac{3+\\sqrt{3}}{4 \\sqrt{2}}<\/script><\/p>\n\n\n\n<p>(ii) cos<sup>2<\/sup> 112<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-30-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-433\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-434\"><span class=\"msubsup\" id=\"MathJax-Span-435\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-436\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-437\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-438\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-439\"><span class=\"mrow\" id=\"MathJax-Span-440\"><span class=\"mo\" id=\"MathJax-Span-441\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-30\">\\frac{1}{2}^{\\circ}<\/script> \u2013 sin<sup>2<\/sup>52<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-31-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-442\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-443\"><span class=\"msubsup\" id=\"MathJax-Span-444\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-445\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-446\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-447\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-448\"><span class=\"mrow\" id=\"MathJax-Span-449\"><span class=\"mo\" id=\"MathJax-Span-450\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-31\">\\frac{1}{2}^{\\circ}<\/script><br> Answer:<br> Use cos<sup>2<\/sup> A \u2013 sin<sup>2<\/sup> B = cos (A + B) cos (A \u2013 B)<br> cos<sup>2<\/sup>112<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-32-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-451\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-452\"><span class=\"msubsup\" id=\"MathJax-Span-453\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-454\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-455\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-456\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-457\"><span class=\"mrow\" id=\"MathJax-Span-458\"><span class=\"mo\" id=\"MathJax-Span-459\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-32\">\\frac{1}{2}^{\\circ}<\/script> \u2013 sin<sup>2<\/sup> 52<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-33-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-460\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-461\"><span class=\"msubsup\" id=\"MathJax-Span-462\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-463\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-464\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-465\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-466\"><span class=\"mrow\" id=\"MathJax-Span-467\"><span class=\"mo\" id=\"MathJax-Span-468\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-33\">\\frac{1}{2}^{\\circ}<\/script><br> = cos [112<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-34-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-469\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-470\"><span class=\"msubsup\" id=\"MathJax-Span-471\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-472\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-473\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-474\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-475\"><span class=\"mrow\" id=\"MathJax-Span-476\"><span class=\"mo\" id=\"MathJax-Span-477\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-34\">\\frac{1}{2}^{\\circ}<\/script> + 52<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-35-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-478\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-479\"><span class=\"msubsup\" id=\"MathJax-Span-480\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-481\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-482\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-483\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-484\"><span class=\"mrow\" id=\"MathJax-Span-485\"><span class=\"mo\" id=\"MathJax-Span-486\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-35\">\\frac{1}{2}^{\\circ}<\/script>]cos[112<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-36-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-487\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-488\"><span class=\"msubsup\" id=\"MathJax-Span-489\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-490\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-491\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-492\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-493\"><span class=\"mrow\" id=\"MathJax-Span-494\"><span class=\"mo\" id=\"MathJax-Span-495\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-36\">\\frac{1}{2}^{\\circ}<\/script> -52<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-37-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-496\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-497\"><span class=\"msubsup\" id=\"MathJax-Span-498\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-499\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-500\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-501\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-502\"><span class=\"mrow\" id=\"MathJax-Span-503\"><span class=\"mo\" id=\"MathJax-Span-504\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-37\">\\frac{1}{2}^{\\circ}<\/script>]<br> = cos 165\u00b0 . cos 60<br> = cos 60\u00b0 cos (180 \u2013 15)<br> = -cos 60\u00b0. cos 15\u00b0<br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-38-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-505\" style=\"width: 11.186em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 9.648em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.866em, 1009.65em, 3.963em, -999.998em); top: -3.166em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-506\"><span class=\"mo\" id=\"MathJax-Span-507\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-508\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-509\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-510\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-511\"><span class=\"mo\" id=\"MathJax-Span-512\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">[<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-513\"><span style=\"display: inline-block; position: relative; width: 1.959em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.311em, 1001.82em, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.93em;\"><span class=\"mrow\" id=\"MathJax-Span-514\"><span class=\"msqrt\" id=\"MathJax-Span-515\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-516\"><span class=\"mn\" id=\"MathJax-Span-517\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-518\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-519\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1001.31em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -0.65em;\"><span class=\"mrow\" id=\"MathJax-Span-520\"><span class=\"mn\" id=\"MathJax-Span-521\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"msqrt\" id=\"MathJax-Span-522\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-523\"><span class=\"mn\" id=\"MathJax-Span-524\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.96em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.959em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-525\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">]<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-526\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mo\" id=\"MathJax-Span-527\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-528\"><span style=\"display: inline-block; position: relative; width: 2.519em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.311em, 1002.33em, 4.336em, -999.998em); top: -4.564em; left: 50%; margin-left: -1.209em;\"><span class=\"mrow\" id=\"MathJax-Span-529\"><span class=\"mo\" id=\"MathJax-Span-530\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">(<\/span><span class=\"msqrt\" id=\"MathJax-Span-531\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-532\"><span class=\"mn\" id=\"MathJax-Span-533\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-534\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-535\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-536\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">)<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1001.31em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -0.65em;\"><span class=\"mrow\" id=\"MathJax-Span-537\"><span class=\"mn\" id=\"MathJax-Span-538\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span class=\"msqrt\" id=\"MathJax-Span-539\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-540\"><span class=\"mn\" id=\"MathJax-Span-541\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1002.52em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.519em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.171em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.808em; border-left: 0px solid; width: 0px; height: 2.219em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-38\">-\\frac{1}{2}\\left[\\frac{\\sqrt{3}+1}{2 \\sqrt{2}}\\right]=-\\frac{(\\sqrt{3}+1)}{4 \\sqrt{2}}<\/script><\/p>\n\n<p>(iii) sin<sup>2<\/sup><span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-39-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-542\" style=\"width: 4.429em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.823em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.4em, 1003.64em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-543\"><span class=\"mrow\" id=\"MathJax-Span-544\"><span class=\"mo\" id=\"MathJax-Span-545\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">[<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-546\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.42em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-547\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-548\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">8<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.51em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.515em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-549\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mfrac\" id=\"MathJax-Span-550\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.51em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.277em;\"><span class=\"mi\" id=\"MathJax-Span-551\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-552\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.65em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.655em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-553\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.425em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.624em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-39\">\\left[\\frac{\\pi}{8}+\\frac{A}{2}\\right]<\/script> \u2013 sin<sup>2<\/sup><span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-40-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-554\" style=\"width: 4.429em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.823em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.4em, 1003.64em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-555\"><span class=\"mrow\" id=\"MathJax-Span-556\"><span class=\"mo\" id=\"MathJax-Span-557\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">[<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-558\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.42em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-559\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-560\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">8<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.51em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.515em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-561\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-562\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.51em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.277em;\"><span class=\"mi\" id=\"MathJax-Span-563\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-564\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.65em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.655em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-565\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.425em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.624em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-40\">\\left[\\frac{\\pi}{8}-\\frac{A}{2}\\right]<\/script><br> Answer:<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7662\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6c-3.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 3\" width=\"319\" height=\"239\" sizes=\"auto, (max-width: 319px) 100vw, 319px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 3\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c)\"><\/p>\n\n\n\n<p>(iv) cos<sup>2<\/sup>52<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-41-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-566\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-567\"><span class=\"msubsup\" id=\"MathJax-Span-568\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-569\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-570\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-571\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-572\"><span class=\"mrow\" id=\"MathJax-Span-573\"><span class=\"mo\" id=\"MathJax-Span-574\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-41\">\\frac{1}{2}^{\\circ}<\/script> \u2013 sin<sup>2<\/sup>22<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-42-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-575\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-576\"><span class=\"msubsup\" id=\"MathJax-Span-577\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-578\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-579\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-580\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-581\"><span class=\"mrow\" id=\"MathJax-Span-582\"><span class=\"mo\" id=\"MathJax-Span-583\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-42\">\\frac{1}{2}^{\\circ}<\/script> (M\u2019 2010, 06 Jun 08)<br> Answer:<br> [\u2235 cos<sup>2<\/sup>A \u2013 sin<sup>2<\/sup>B = cos (A + B) cos (A \u2013 B)]<br> cos<sup>2<\/sup>52<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-43-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-584\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-585\"><span class=\"msubsup\" id=\"MathJax-Span-586\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-587\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-588\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-589\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-590\"><span class=\"mrow\" id=\"MathJax-Span-591\"><span class=\"mo\" id=\"MathJax-Span-592\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-43\">\\frac{1}{2}^{\\circ}<\/script> \u2013 sin<sup>2<\/sup>22<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-44-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-593\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-594\"><span class=\"msubsup\" id=\"MathJax-Span-595\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-596\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-597\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-598\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-599\"><span class=\"mrow\" id=\"MathJax-Span-600\"><span class=\"mo\" id=\"MathJax-Span-601\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-44\">\\frac{1}{2}^{\\circ}<\/script><br> = cos[52<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-45-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-602\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-603\"><span class=\"msubsup\" id=\"MathJax-Span-604\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-605\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-606\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-607\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-608\"><span class=\"mrow\" id=\"MathJax-Span-609\"><span class=\"mo\" id=\"MathJax-Span-610\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-45\">\\frac{1}{2}^{\\circ}<\/script> + 22<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-46-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-611\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-612\"><span class=\"msubsup\" id=\"MathJax-Span-613\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-614\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-615\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-616\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-617\"><span class=\"mrow\" id=\"MathJax-Span-618\"><span class=\"mo\" id=\"MathJax-Span-619\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-46\">\\frac{1}{2}^{\\circ}<\/script>]cos[52<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-47-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-620\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-621\"><span class=\"msubsup\" id=\"MathJax-Span-622\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-623\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-624\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-625\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-626\"><span class=\"mrow\" id=\"MathJax-Span-627\"><span class=\"mo\" id=\"MathJax-Span-628\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-47\">\\frac{1}{2}^{\\circ}<\/script> \u2013 22<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-48-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-629\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-630\"><span class=\"msubsup\" id=\"MathJax-Span-631\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.7em, 4.522em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mfrac\" id=\"MathJax-Span-632\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-633\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-634\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.611em; left: 0.701em;\"><span class=\"texatom\" id=\"MathJax-Span-635\"><span class=\"mrow\" id=\"MathJax-Span-636\"><span class=\"mo\" id=\"MathJax-Span-637\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-48\">\\frac{1}{2}^{\\circ}<\/script>]<br> = cos 75\u00b0 cos 30\u00b0<br> = cos 30\u00b0 cos(90 \u2013 15)<br> = cos 30\u00b0 sin 15\u00b0<br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-49-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-638\" style=\"width: 9.741em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 8.39em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.866em, 1008.39em, 3.963em, -999.998em); top: -3.166em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-639\"><span class=\"mfrac\" id=\"MathJax-Span-640\"><span style=\"display: inline-block; position: relative; width: 1.074em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.311em, 1000.93em, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.464em;\"><span class=\"msqrt\" id=\"MathJax-Span-641\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-642\"><span class=\"mn\" id=\"MathJax-Span-643\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-644\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.07em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.074em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-645\"><span class=\"mo\" id=\"MathJax-Span-646\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-647\"><span style=\"display: inline-block; position: relative; width: 1.959em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.311em, 1001.82em, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.93em;\"><span class=\"mrow\" id=\"MathJax-Span-648\"><span class=\"msqrt\" id=\"MathJax-Span-649\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-650\"><span class=\"mn\" id=\"MathJax-Span-651\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-652\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-653\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1001.31em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -0.65em;\"><span class=\"mrow\" id=\"MathJax-Span-654\"><span class=\"mn\" id=\"MathJax-Span-655\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"msqrt\" id=\"MathJax-Span-656\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-657\"><span class=\"mn\" id=\"MathJax-Span-658\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.96em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.959em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-659\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">)<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-660\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-661\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.959em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.311em, 1001.87em, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.93em;\"><span class=\"mrow\" id=\"MathJax-Span-662\"><span class=\"mn\" id=\"MathJax-Span-663\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"mo\" id=\"MathJax-Span-664\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"msqrt\" id=\"MathJax-Span-665\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-666\"><span class=\"mn\" id=\"MathJax-Span-667\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1001.31em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -0.65em;\"><span class=\"mrow\" id=\"MathJax-Span-668\"><span class=\"mn\" id=\"MathJax-Span-669\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span class=\"msqrt\" id=\"MathJax-Span-670\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-671\"><span class=\"mn\" id=\"MathJax-Span-672\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.96em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.959em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.171em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.808em; border-left: 0px solid; width: 0px; height: 2.219em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-49\">\\frac{\\sqrt{3}}{2}\\left(\\frac{\\sqrt{3}-1}{2 \\sqrt{2}}\\right)=\\frac{3-\\sqrt{3}}{4 \\sqrt{2}}<\/script><\/p>\n\n\n\n<h3>Question 6.<br> Find the minimum and maximum values of<br> (i) 3 cos x + 4 sin x<\/h3>\n\n<p>Answer:<br> Recall for a cos x + b sin x + c<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7661\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6c-4.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 4\" width=\"317\" height=\"163\" sizes=\"auto, (max-width: 317px) 100vw, 317px\" data-pin-nopin=\"true\"><\/p>\n\n<p>(ii) sin 2x \u2013 cos 2x<br> Answer:<br> a = 1, b = -1, c = 0<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7660\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6c-5.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 5\" width=\"319\" height=\"153\" sizes=\"auto, (max-width: 319px) 100vw, 319px\" data-pin-nopin=\"true\"><\/p>\n\n<h3>Question 7.<br> Find the range of<br> (i) 7 cos x \u2013 24 sin x + 5<\/h3>\n\n<p>Answer:<br> a = -24, b = 7, c = 5<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7659\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6c-6.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 6\" width=\"319\" height=\"183\" sizes=\"auto, (max-width: 319px) 100vw, 319px\" data-pin-nopin=\"true\"><\/p>\n\n<p>(ii) 13 cos x + 3\u221a3 sin x \u2013 4<br> Answer:<br> a = 3\u221a3, b = 13, c = -4<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7658\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6c-7.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 7\" width=\"310\" height=\"192\" sizes=\"auto, (max-width: 310px) 100vw, 310px\" data-pin-nopin=\"true\"><\/p>\n\n<h3>II.<br> Question 1.<br> (i)If cos \u03b1 = \u2013<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-50-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-673\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-674\"><span class=\"mfrac\" id=\"MathJax-Span-675\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-676\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-677\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-50\">\\frac{3}{5}<\/script> and sin \u03b2 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-51-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-678\" style=\"width: 1.307em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.121em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.12em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-679\"><span class=\"mfrac\" id=\"MathJax-Span-680\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-681\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">7<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.65em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-682\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">25<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.84em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.841em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-51\">\\frac{7}{25}<\/script> where <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-52-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-683\" style=\"width: 0.934em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.795em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.447em, 1000.79em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-684\"><span class=\"mfrac\" id=\"MathJax-Span-685\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.42em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-686\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-687\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.51em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.515em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-52\">\\frac{\\pi}{2}<\/script> &lt; \u03b1 &lt; \u03c0 and 0 &lt; \u03b2 &lt; <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-53-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-688\" style=\"width: 0.934em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.795em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.447em, 1000.79em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-689\"><span class=\"mfrac\" id=\"MathJax-Span-690\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.42em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-691\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-692\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.51em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.515em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-53\">\\frac{\\pi}{2}<\/script>, then find the values of tan(\u03b1 + \u03b2) and sin(\u03b1 + \u03b2).<\/h3>\n\n<p>Answer:<br> cos \u03b1 = \u2013<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-54-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-693\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-694\"><span class=\"mfrac\" id=\"MathJax-Span-695\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-696\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-697\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-54\">\\frac{3}{5}<\/script>, and \u03b1 lies in second quadrant<br> sin \u03b1 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-55-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-698\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-699\"><span class=\"mfrac\" id=\"MathJax-Span-700\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-701\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-702\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-55\">\\frac{4}{5}<\/script> \u2234 tan \u03b1 = \u2013<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-56-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-703\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-704\"><span class=\"mfrac\" id=\"MathJax-Span-705\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-706\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-707\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-56\">\\frac{4}{5}<\/script><br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7657\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6c-8.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 8\" width=\"316\" height=\"390\" sizes=\"auto, (max-width: 316px) 100vw, 316px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 8\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c)\"><\/p>\n\n\n\n<p>(ii) If 0 &lt; A &lt; B &lt; <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-57-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-708\" style=\"width: 0.934em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.795em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.447em, 1000.79em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-709\"><span class=\"mfrac\" id=\"MathJax-Span-710\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.42em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-711\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-712\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.51em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.515em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-57\">\\frac{\\pi}{4}<\/script> and sin(A + B) = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-58-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-713\" style=\"width: 1.307em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.121em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.12em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-714\"><span class=\"mfrac\" id=\"MathJax-Span-715\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.7em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-716\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">24<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.65em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-717\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">25<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.84em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.841em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-58\">\\frac{24}{25}<\/script> and cos (A \u2013 B) = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-59-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-718\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-719\"><span class=\"mfrac\" id=\"MathJax-Span-720\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-721\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-722\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-59\">\\frac{4}{5}<\/script>, then find the value of tan 2A. (March 2015-T.S)<br>\n\nAnswer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7656\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6c-9.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 9\" width=\"320\" height=\"510\" sizes=\"auto, (max-width: 320px) 100vw, 320px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 9\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c)\"><\/p>\n\n<p>(iii) If A + B, A are acute angles such that sin (A + B) = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-60-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-723\" style=\"width: 1.307em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.121em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.12em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-724\"><span class=\"mfrac\" id=\"MathJax-Span-725\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.7em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-726\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">24<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.65em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-727\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">25<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.84em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.841em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-60\">\\frac{24}{25}<\/script> and tan A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-61-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-728\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-729\"><span class=\"mfrac\" id=\"MathJax-Span-730\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-731\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-732\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-61\">\\frac{3}{5}<\/script>, then find the value of cos B.<br> Answer:<br> A + B, A are acute angles \u21d2 B is also acute.<br> Given sin (A + B) = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-62-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-733\" style=\"width: 1.307em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.121em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.12em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-734\"><span class=\"mfrac\" id=\"MathJax-Span-735\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.7em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-736\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">24<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.65em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-737\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">25<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.84em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.841em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-62\">\\frac{24}{25}<\/script>, we have<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7655\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6c-10.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 10\" width=\"296\" height=\"511\" sizes=\"auto, (max-width: 296px) 100vw, 296px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 10\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c)\"><\/p>\n\n<p>(iv) If tan \u03b1 \u2013 tan \u03b2 = m, and cot \u03b1 \u2013 cot \u03b2 = n then prove that cot(\u03b1 \u2013 \u03b2) = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-63-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-738\" style=\"width: 3.73em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.218em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.307em, 1003.22em, 2.798em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-739\"><span class=\"mfrac\" id=\"MathJax-Span-740\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-741\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.544em, 1000.75em, 4.15em, -999.998em); top: -3.679em; left: 50%; margin-left: -0.37em;\"><span class=\"texatom\" id=\"MathJax-Span-742\"><span class=\"mrow\" id=\"MathJax-Span-743\"><span class=\"mtext\" id=\"MathJax-Span-744\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">&nbsp;<\/span><span class=\"mi\" id=\"MathJax-Span-745\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">m<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-746\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-747\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-748\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.544em, 1000.38em, 4.15em, -999.998em); top: -3.679em; left: 50%; margin-left: -0.184em;\"><span class=\"texatom\" id=\"MathJax-Span-749\"><span class=\"mrow\" id=\"MathJax-Span-750\"><span class=\"mi\" id=\"MathJax-Span-751\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">n<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.51em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.515em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.43em; border-left: 0px solid; width: 0px; height: 1.516em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-63\">\\frac{1}{\\mathrm{~m}}-\\frac{1}{\\mathrm{n}}<\/script>.<br>\n\nAnswer:<br> We have tan \u03b1 \u2013 tan \u03b2 = m<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7654\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6c-11.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 11\" width=\"313\" height=\"389\" sizes=\"auto, (max-width: 313px) 100vw, 313px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 11\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c)\"><\/p>\n\n\n\n<p>(v) If tan (\u03b1 \u2013 \u03b2) = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-64-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-752\" style=\"width: 1.307em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.121em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.12em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-753\"><span class=\"mfrac\" id=\"MathJax-Span-754\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-755\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">7<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.7em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-756\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">24<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.84em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.841em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-64\">\\frac{7}{24}<\/script> and tan \u03b1 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-65-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-757\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-758\"><span class=\"mfrac\" id=\"MathJax-Span-759\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-760\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-761\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-65\">\\frac{4}{3}<\/script> where \u03b1 and \u03b2 are in the first quadrant prove that \u03b1 + \u03b2 = \u03c0\/2.<br> Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7653\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6c-12.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 12\" width=\"272\" height=\"564\" sizes=\"auto, (max-width: 272px) 100vw, 272px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 12\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c)\"><\/p>\n\n<h3>Question 2.<br> i) Find the expansion of sin (A + B \u2013 C).<\/h3>\n\n<p>Answer:<br> 0 sin (A + B \u2013 C) = Sin [ (A + B) \u2013 C]<br> = sin (A + B) cos C \u2013 cos (A + B) sin C] \u2013 (sin A cos B + cos A sin B) cos C \u2013 (cos A cos B \u2013 sin A sin B] sin C<br> = sin A cos B cos C + cos A sin B cos C \u2013 cos A cos B sin C \u2013 sin A sin B sin C<\/p>\n\n<p>ii) Find the expansion of cos (A \u2013 B \u2013 C).<br> Answer:<br> cos (A \u2013 B \u2013 C) = cos [(A \u2013 B) \u2013 C]<br> = cos (A \u2013 B) cos C + sin (A \u2013 B) sin C<br> = (cos A cos B + sin A sin B) cos C + (sin A cos B \u2013 cos A sin B] sin C<br> = cos A cos B cos C + sin A sin B cos C + sin A cos B sin C \u2013 cos A sin B sin C<\/p>\n\n<p>iii) In a \u0394ABC, A is obtuse. If sin A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-66-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-762\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-763\"><span class=\"mfrac\" id=\"MathJax-Span-764\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-765\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-766\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-66\">\\frac{3}{5}<\/script> and sin B = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-67-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-767\" style=\"width: 1.307em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.121em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.12em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-768\"><span class=\"mfrac\" id=\"MathJax-Span-769\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-770\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.7em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-771\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">13<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.84em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.841em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-67\">\\frac{5}{13}<\/script>, then show that sin C = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-68-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-772\" style=\"width: 1.307em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.121em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.12em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-773\"><span class=\"mfrac\" id=\"MathJax-Span-774\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.7em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-775\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">16<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.65em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-776\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">65<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.84em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.841em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-68\">\\frac{16}{65}<\/script><br> Answer:<br> Given, A + B + C = 180\u00b0<br> \u21d2 A + B = 180\u00b0 \u2013 C<br> \u2234 sin (A + B) = sin (180\u00b0 \u2013 C)<br> = sin C \u2026\u2026\u2026\u2026\u2026\u2026..(1)<br> \u2234 A is obtuse angle and A lies in II quadrant.<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7652\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6c-13.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 13\" width=\"309\" height=\"230\" sizes=\"auto, (max-width: 309px) 100vw, 309px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 13\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c)\"><\/p>\n\n<p>iv) If <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-69-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-777\" style=\"width: 7.225em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.2em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.074em, 1006.2em, 3.078em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-778\"><span class=\"mfrac\" id=\"MathJax-Span-779\"><span style=\"display: inline-block; position: relative; width: 2.938em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1002.75em, 4.336em, -999.998em); top: -4.564em; left: 50%; margin-left: -1.396em;\"><span class=\"mrow\" id=\"MathJax-Span-780\"><span class=\"mi\" id=\"MathJax-Span-781\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-782\" style=\"font-size: 70.7%;\"><\/span><span class=\"mo\" id=\"MathJax-Span-783\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">(<\/span><span class=\"mi\" id=\"MathJax-Span-784\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03b1<\/span><span class=\"mo\" id=\"MathJax-Span-785\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-786\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03b2<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-787\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">)<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1002.75em, 4.336em, -999.998em); top: -3.586em; left: 50%; margin-left: -1.396em;\"><span class=\"mrow\" id=\"MathJax-Span-788\"><span class=\"mi\" id=\"MathJax-Span-789\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-790\" style=\"font-size: 70.7%;\"><\/span><span class=\"mo\" id=\"MathJax-Span-791\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">(<\/span><span class=\"mi\" id=\"MathJax-Span-792\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03b1<\/span><span class=\"mo\" id=\"MathJax-Span-793\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-794\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03b2<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-795\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">)<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1002.94em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.938em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-796\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-797\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.354em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1001.21em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -0.603em;\"><span class=\"mrow\" id=\"MathJax-Span-798\"><span class=\"mi\" id=\"MathJax-Span-799\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-800\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-801\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">b<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.21em, 4.196em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.603em;\"><span class=\"mrow\" id=\"MathJax-Span-802\"><span class=\"mi\" id=\"MathJax-Span-803\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-804\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-805\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">b<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.35em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.354em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.754em; border-left: 0px solid; width: 0px; height: 2.111em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-69\">\\frac{\\sin (\\alpha+\\beta)}{\\sin (\\alpha-\\beta)}=\\frac{a+b}{a-b}<\/script> then prove that a tan \u03b2 = b tan \u03b1.<br> Answer:<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7651\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6c-14.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 14\" width=\"318\" height=\"602\" sizes=\"auto, (max-width: 318px) 100vw, 318px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c) 14\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(c)\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>I. Question 1. Simplify the following (i) cos 100\u00b0 . cos 40\u00b0 + sin 100\u00b0 . sin 40\u00b0 Answer: Use cos A. cos B + sin A sin B = cos (A \u2013 B) \u2234 cos 100\u00b0 . cos 40\u00b0 + sin 100\u00b0.sin 40\u00b0 = cos (100\u00b0 \u2013 40\u00b0) = cos 60\u00b0 = 12 = [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":3355,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1544,15,38],"tags":[2088,2085,2090,2093,2087,2091,2083,2089,2082,2092,2095,1557,2086,2094,2084],"class_list":{"0":"post-3303","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-study-material-tg-inter","8":"category-telangana","9":"category-tg-inter","10":"tag-exercise-6c-trigonometry-problems-with-solutions","11":"tag-inter-1st-year-maths-1a-trigonometry-solutions","12":"tag-intermediate-1st-year-trigonometry-exercise-6c","13":"tag-maths-1a-trigonometry-exercise-6c-solutions-in-telugu","14":"tag-trigonometric-ratios-transformations-maths-1a","15":"tag-trigonometric-ratios-upto-transformations-exercise-6c-explained","16":"tag-trigonometric-ratios-upto-transformations-solutions","17":"tag-ts-board-maths-1a-trigonometric-ratios-solutions","18":"tag-ts-inter-1st-year-maths-1a-exercise-6c-solutions","19":"tag-ts-inter-1st-year-maths-1a-exercise-6c-step-by-step-solutions","20":"tag-ts-inter-1st-year-maths-1a-solved-examples-pdf","21":"tag-ts-inter-1st-year-maths-1a-textbook-solutions","22":"tag-ts-inter-maths-1a-chapter-6c-answers","23":"tag-ts-intermediate-maths-1a-trigonometry-exam-preparation","24":"tag-ts-intermediate-maths-1a-trigonometry-exercise-6c"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>TS Inter 1st Year Maths 1A Trigonometric Ratios upto Transformations Solutions Exercise 6(c)<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6c\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"TS Inter 1st Year Maths 1A Trigonometric Ratios upto Transformations Solutions Exercise 6(c)\" \/>\n<meta property=\"og:description\" content=\"I. Question 1. Simplify the following (i) cos 100\u00b0 . cos 40\u00b0 + sin 100\u00b0 . sin 40\u00b0 Answer: Use cos A. cos B + sin A sin B = cos (A \u2013 B) \u2234 cos 100\u00b0 . cos 40\u00b0 + sin 100\u00b0.sin 40\u00b0 = cos (100\u00b0 \u2013 40\u00b0) = cos 60\u00b0 = 12 = [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6c\/\" \/>\n<meta property=\"og:site_name\" content=\"Manabadi Boards\" \/>\n<meta property=\"article:published_time\" content=\"2026-01-28T12:30:42+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2026-01-30T09:06:58+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114617\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6c.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"900\" \/>\n\t<meta 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\/ Yoast SEO plugin. -->","yoast_head_json":{"title":"TS Inter 1st Year Maths 1A Trigonometric Ratios upto Transformations Solutions Exercise 6(c)","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6c\/","og_locale":"en_US","og_type":"article","og_title":"TS Inter 1st Year Maths 1A Trigonometric Ratios upto Transformations Solutions Exercise 6(c)","og_description":"I. Question 1. Simplify the following (i) cos 100\u00b0 . cos 40\u00b0 + sin 100\u00b0 . sin 40\u00b0 Answer: Use cos A. cos B + sin A sin B = cos (A \u2013 B) \u2234 cos 100\u00b0 . cos 40\u00b0 + sin 100\u00b0.sin 40\u00b0 = cos (100\u00b0 \u2013 40\u00b0) = cos 60\u00b0 = 12 = [&hellip;]","og_url":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6c\/","og_site_name":"Manabadi Boards","article_published_time":"2026-01-28T12:30:42+00:00","article_modified_time":"2026-01-30T09:06:58+00:00","og_image":[{"width":900,"height":600,"url":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114617\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6c.jpg","type":"image\/jpeg"}],"author":"Junaid","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Junaid","Est. reading time":"10 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