{"id":3369,"date":"2026-01-31T12:09:37","date_gmt":"2026-01-31T06:39:37","guid":{"rendered":"https:\/\/www.manabadi.co.in\/boards\/?p=3369"},"modified":"2026-02-02T13:00:18","modified_gmt":"2026-02-02T07:30:18","slug":"ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6d","status":"publish","type":"post","link":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6d\/","title":{"rendered":"TS Inter 1st Year Maths 1A Trigonometric Ratios upto Transformations Solutions Exercise 6(d)"},"content":{"rendered":"\n<h3>I.<br> Question 1.<br> Simplify<br> (i) <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: 3.637em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.124em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1003.12em, 2.891em, -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: 2.845em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1001.73em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.836em;\"><span class=\"mrow\" id=\"MathJax-Span-4\"><span class=\"mi\" id=\"MathJax-Span-5\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-6\" style=\"font-size: 70.7%;\"><\/span><span class=\"mn\" id=\"MathJax-Span-7\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-8\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03b8<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1002.71em, 4.196em, -999.998em); top: -3.632em; left: 50%; margin-left: -1.349em;\"><span class=\"mrow\" id=\"MathJax-Span-9\"><span class=\"mn\" id=\"MathJax-Span-10\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-11\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-12\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-13\" style=\"font-size: 70.7%;\"><\/span><span class=\"mn\" id=\"MathJax-Span-14\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-15\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03b8<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1002.85em, 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: 2.845em; 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: 1.678em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-1\">\\frac{\\sin 2 \\theta}{1+\\cos 2 \\theta}<\/script><\/h3>\n\n<p>Answer:<br> <img decoding=\"async\" class=\"alignnone size-full wp-image-7827\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-1.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 1\" width=\"309\" height=\"107\" sizes=\"(max-width: 309px) 100vw, 309px\" data-pin-nopin=\"true\"><\/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-16\" style=\"width: 5.454em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.709em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1004.71em, 2.891em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-17\"><span class=\"mfrac\" id=\"MathJax-Span-18\"><span style=\"display: inline-block; position: relative; width: 4.429em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1004.29em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -2.141em;\"><span class=\"mrow\" id=\"MathJax-Span-19\"><span class=\"mn\" id=\"MathJax-Span-20\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-21\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-22\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-23\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">\u03b8<\/span><span class=\"mo\" id=\"MathJax-Span-24\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-25\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-26\" style=\"font-size: 70.7%;\"><\/span><span class=\"mn\" id=\"MathJax-Span-27\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-28\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03b8<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1004.15em, 4.196em, -999.998em); top: -3.632em; left: 50%; margin-left: -2.095em;\"><span class=\"mrow\" id=\"MathJax-Span-29\"><span class=\"mn\" id=\"MathJax-Span-30\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-31\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-32\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-33\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">\u03b8<\/span><span class=\"mo\" id=\"MathJax-Span-34\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-35\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-36\" style=\"font-size: 70.7%;\"><\/span><span class=\"mn\" id=\"MathJax-Span-37\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-38\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03b8<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1004.43em, 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: 4.429em; 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{3 \\cos \\theta+\\cos 3 \\theta}{3 \\sin \\theta-\\sin 3 \\theta}<\/script><br>\n\nAnswer:<br> <img decoding=\"async\" class=\"alignnone size-full wp-image-7826\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-2.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 2\" width=\"237\" height=\"186\" data-pin-nopin=\"true\"><\/p>\n\n\n\n<h3>Question 2.<br> Evaluate the following<br> (i) 6sin 20\u00b0 \u2013 8 sin<sup>3<\/sup> 20\u00b0<\/h3>\n\n<p>Answer:<br> 2(3 sin 20\u00b0 \u2013 4sin<sup>3<\/sup> 20\u00b0) (Formula)<br> = 2.sin (3 \u00d7 20\u00b0) = 2 sin 60\u00b0<br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-3-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-39\" style=\"width: 1.959em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.68em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.027em, 1001.68em, 2.845em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-40\"><span class=\"mfrac\" id=\"MathJax-Span-41\"><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, 1001.31em, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.65em;\"><span class=\"mrow\" id=\"MathJax-Span-42\"><span class=\"mn\" id=\"MathJax-Span-43\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"msqrt\" id=\"MathJax-Span-44\"><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-45\"><span class=\"mn\" id=\"MathJax-Span-46\" 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(3.404em, 1000.89em, 4.196em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.464em;\"><span class=\"mrow\" id=\"MathJax-Span-47\"><span class=\"mo\" id=\"MathJax-Span-48\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-49\" 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(0.888em, 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><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: 1.895em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-3\">\\frac{2 \\sqrt{3}}{-2}<\/script> = \u221a3<\/p>\n\n<p>(ii) cos<sup>2<\/sup>72\u00b0 \u2013 sin<sup>2<\/sup>54\u00b0<br> Answer:<br> cos<sup>2<\/sup>72\u00b0 \u2013 sin<sup>2<\/sup>54\u00b0<br> = sin<sup>2<\/sup>18\u00b0 \u2013 cos<sup>2<\/sup>36\u00b0<br> <img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-7825\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-3.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 3\" width=\"292\" height=\"220\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 3\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d)\"><\/p>\n\n<p>(iii) sin<sup>2<\/sup>42\u00b0 \u2013 sin<sup>2<\/sup>12\u00b0<br> Answer:<br> sin<sup>2<\/sup>42\u00b0 \u2013 sin<sup>2<\/sup>12\u00b0 (Formula)<br> sin (42\u00b0 + 12\u00b0) sin (42\u00b0 \u2013 12\u00b0)<br> = sin 54\u00b0 sin 30\u00b0<br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-4-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-50\" style=\"width: 9.275em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 7.971em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.866em, 1007.97em, 3.963em, -999.998em); top: -3.166em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-51\"><span class=\"mrow\" id=\"MathJax-Span-52\"><span class=\"mo\" id=\"MathJax-Span-53\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-54\"><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-55\"><span class=\"msqrt\" id=\"MathJax-Span-56\"><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-57\"><span class=\"mn\" id=\"MathJax-Span-58\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/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-59\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-60\" 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(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-61\" 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, 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-62\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">)<\/span><\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-63\" style=\"padding-left: 0.189em;\"><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-64\" 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-65\" 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-66\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-67\" 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, 1001.82em, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.93em;\"><span class=\"mrow\" id=\"MathJax-Span-68\"><span class=\"msqrt\" id=\"MathJax-Span-69\"><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-70\"><span class=\"mn\" id=\"MathJax-Span-71\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/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-72\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-73\" 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(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-74\" 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, 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-4\">\\left(\\frac{\\sqrt{5}+1}{4}\\right) \\frac{1}{2}=\\frac{\\sqrt{5}+1}{8}<\/script><\/p>\n\n\n\n<h3>Question 3.<br> (i) Express <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-5-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-75\" 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-76\"><span class=\"mfrac\" id=\"MathJax-Span-77\"><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(3.357em, 1001.73em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.836em;\"><span class=\"mrow\" id=\"MathJax-Span-78\"><span class=\"mi\" id=\"MathJax-Span-79\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-80\" style=\"font-size: 70.7%;\"><\/span><span class=\"mn\" id=\"MathJax-Span-81\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">4<\/span><span class=\"mi\" id=\"MathJax-Span-82\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03b8<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.35em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.697em;\"><span class=\"mrow\" id=\"MathJax-Span-83\"><span class=\"mi\" id=\"MathJax-Span-84\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-85\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-86\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">\u03b8<\/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: 1.624em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-5\">\\frac{\\sin 4 \\theta}{\\sin \\theta}<\/script> in terms of cos3 \u03b8 and cos \u03b8.<\/h3>\n\n<p>Answer:<br> sin4\u03b8 = sin (3\u03b8 + \u03b8) = sin 3\u03b8 cos \u03b8 +cos 3\u03b8 sin \u03b8<br> = (3 sin \u03b8 \u2013 i sin 3\u03b8) cos \u03b8 + (4 cos 3\u03b8 \u2013 3 cos \u03b8) sin \u03b8<br> = 3 sin \u03b8 cos \u03b8 \u2013 4 sin 3\u03b8 cos \u03b8 + 4 cos 3\u03b8 sin \u03b8 \u2013 3 cos \u03b8 sin \u03b8<br> = 4 cos 3\u03b8 sin \u03b8 \u2013 4 sin 3\u03b8 cos \u03b8<br> = sin \u03b8(1 cos 3\u03b8 \u2013 4 sin 2\u03b8 cos \u03b8)<br> = sin \u03b8 [4 cos 3\u03b8 \u2013 4 sin 2\u03b8 cos \u03b8]<br> \u2234 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-6-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-87\" style=\"width: 14.541em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 12.537em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(0.934em, 1012.54em, 2.891em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-88\"><span class=\"mfrac\" id=\"MathJax-Span-89\"><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(3.357em, 1001.73em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.836em;\"><span class=\"mrow\" id=\"MathJax-Span-90\"><span class=\"mi\" id=\"MathJax-Span-91\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-92\" style=\"font-size: 70.7%;\"><\/span><span class=\"mn\" id=\"MathJax-Span-93\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">4<\/span><span class=\"mi\" id=\"MathJax-Span-94\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03b8<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.35em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.697em;\"><span class=\"mrow\" id=\"MathJax-Span-95\"><span class=\"mi\" id=\"MathJax-Span-96\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-97\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-98\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">\u03b8<\/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 class=\"mo\" id=\"MathJax-Span-99\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-100\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 8.809em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.264em, 1008.58em, 4.383em, -999.998em); top: -4.658em; left: 50%; margin-left: -4.331em;\"><span class=\"mrow\" id=\"MathJax-Span-101\"><span class=\"mi\" id=\"MathJax-Span-102\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-103\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-104\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">\u03b8<\/span><span class=\"mrow\" id=\"MathJax-Span-105\"><span class=\"mo\" id=\"MathJax-Span-106\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size1;\">(<\/span><\/span><\/span><span class=\"mn\" id=\"MathJax-Span-107\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span class=\"msubsup\" id=\"MathJax-Span-108\" 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-109\" 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-110\" style=\"font-size: 50%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-111\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-112\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">\u03b8<\/span><span class=\"mo\" id=\"MathJax-Span-113\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-114\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span class=\"msubsup\" id=\"MathJax-Span-115\" style=\"padding-left: 0.189em;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.84em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-116\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.285em; left: 0.888em;\"><span class=\"mn\" id=\"MathJax-Span-117\" 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-118\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-119\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">\u03b8<\/span><span class=\"mi\" id=\"MathJax-Span-120\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-121\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-122\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">\u03b8<\/span><span class=\"mo\" id=\"MathJax-Span-123\" 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, 1001.35em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.697em;\"><span class=\"mrow\" id=\"MathJax-Span-124\"><span class=\"mi\" id=\"MathJax-Span-125\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-126\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-127\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">\u03b8<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1008.81em, 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: 8.809em; 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-6\">\\frac{\\sin 4 \\theta}{\\sin \\theta}=\\frac{\\sin \\theta\\left(4 \\cos ^3 \\theta-4 \\sin ^2 \\theta \\cos \\theta\\right)}{\\sin \\theta}<\/script><br> = 4 cos 3\u03b8 \u2013 4 sin 2\u03b8 cos \u03b8<br> = 4 cos 3\u03b8 \u2013 4 (1- cos 2\u03b8) cos \u03b8<br> = 8 cos 3\u03b8 \u2013 4 cos \u03b8<\/p>\n\n<p>(ii) Express cos<sup>6<\/sup> A + sin<sup>6<\/sup> A in terms of sin 2A.<br> Answer:<br> cos<sup>6<\/sup>A + sin<sup>6<\/sup>A = (cos<sup>2<\/sup>A + sin<sup>2<\/sup>A)<sup>3<\/sup><br> = (cos<sup>2<\/sup>A + sin<sup>2<\/sup>A)<sup>3<\/sup> \u2013 3 cos<sup>2<\/sup>A sin<sup>2<\/sup>A (cos<sup>2<\/sup>A + sin<sup>2<\/sup> A)<br> = 1 \u2013 3 cos<sup>2<\/sup>A sin<sup>2<\/sup>A \u2026\u2026\u2026\u2026\u2026(1)<br> = 1 \u2013 <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: 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-129\"><span class=\"mfrac\" id=\"MathJax-Span-130\"><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-131\" 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-132\" 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, 1000.47em, 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-7\">\\frac{3}{4}<\/script> (4 cos<sup>2<\/sup> A sin<sup>2<\/sup> A)<br> = 1 \u2013 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-8-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-133\" 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-134\"><span class=\"mfrac\" id=\"MathJax-Span-135\"><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-136\" 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-137\" 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, 1000.47em, 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-8\">\\frac{3}{4}<\/script>sin<sup>2<\/sup>2A<\/p>\n\n<p>(iii) Express <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-9-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-138\" style=\"width: 5.408em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.662em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1004.66em, 2.891em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-139\"><span class=\"mfrac\" id=\"MathJax-Span-140\"><span style=\"display: inline-block; position: relative; width: 4.383em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1004.24em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -2.141em;\"><span class=\"mrow\" id=\"MathJax-Span-141\"><span class=\"mn\" id=\"MathJax-Span-142\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-143\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-144\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/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;\">\u03b8<\/span><span class=\"mo\" id=\"MathJax-Span-147\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><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;\">\u03b8<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1004.24em, 4.196em, -999.998em); top: -3.632em; left: 50%; margin-left: -2.141em;\"><span class=\"mrow\" id=\"MathJax-Span-151\"><span class=\"mn\" id=\"MathJax-Span-152\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-153\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-154\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-155\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-156\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">\u03b8<\/span><span class=\"mo\" id=\"MathJax-Span-157\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-158\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-159\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-160\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">\u03b8<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1004.38em, 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: 4.383em; 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-9\">\\frac{1-\\cos \\theta+\\sin \\theta}{1+\\cos \\theta+\\sin \\theta}<\/script> in terms of tan <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-10-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-161\" 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-162\"><span class=\"mfrac\" id=\"MathJax-Span-163\"><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(3.357em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-164\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03b8<\/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-165\" 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-10\">\\frac{\\theta}{2}<\/script><br> Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7824\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-4.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 4\" width=\"262\" height=\"292\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 4\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d)\"><\/p>\n\n\n\n<h3>Question 4.<br> (i) If sin \u03b1 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-11-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-166\" 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-167\"><span class=\"mfrac\" id=\"MathJax-Span-168\"><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-169\" 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-170\" 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-11\">\\frac{3}{5}<\/script>, where <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-12-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-171\" 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-172\"><span class=\"mfrac\" id=\"MathJax-Span-173\"><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-174\" 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-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, 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-12\">\\frac{\\pi}{2}<\/script> &lt; \u03b1 &lt; \u03c0, evaluate cos 3\u03b1 and tan 2\u03b1. (March 2015-T.S)<\/h3> <p>Answer:<br> since <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-13-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-176\" 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-177\"><span class=\"mfrac\" id=\"MathJax-Span-178\"><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-179\" 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-180\" 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-13\">\\frac{\\pi}{2}<\/script> &lt; \u03b1 &lt; \u03c0, a lies in second quadrant and given sin \u03b1 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-14-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-181\" 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-182\"><span class=\"mfrac\" id=\"MathJax-Span-183\"><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-184\" 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-185\" 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-14\">\\frac{3}{5}<\/script> we have cos \u03b1 = \u2013<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-15-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-186\" 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-187\"><span class=\"mfrac\" id=\"MathJax-Span-188\"><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-189\" 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-190\" 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-15\">\\frac{4}{5}<\/script><br> cos 3\u03b1 = 4 cos<sup>3<\/sup> \u03b1 \u2013 3 cos \u03b1<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7823\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-5.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 5\" width=\"292\" height=\"356\" sizes=\"auto, (max-width: 292px) 100vw, 292px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 5\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d)\"><\/p>\n\n<p>(ii) If cos A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-16-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-191\" 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-192\"><span class=\"mfrac\" id=\"MathJax-Span-193\"><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-194\" 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-195\" 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-16\">\\frac{7}{25}<\/script> and <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-17-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-196\" 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-197\"><span class=\"mfrac\" id=\"MathJax-Span-198\"><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-199\"><span class=\"mn\" id=\"MathJax-Span-200\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-201\" 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-202\" 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, 1000.89em, 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-17\">\\frac{3 \\pi}{2}<\/script> &lt; A &lt; 2\u03c0, then find the value of cot <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-18-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-203\" style=\"width: 1.074em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.93em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-204\"><span class=\"mfrac\" id=\"MathJax-Span-205\"><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-206\" 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-207\" 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><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{A}{2}<\/script>.<br> Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7822\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-6.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 6\" width=\"328\" height=\"348\" sizes=\"auto, (max-width: 328px) 100vw, 328px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 6\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d)\"><\/p>\n\n<p>(iii) If 0 &lt; \u03b8 &lt; <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-19-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-208\" 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-209\"><span class=\"mfrac\" id=\"MathJax-Span-210\"><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-211\" 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-212\" 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><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-19\">\\frac{\\pi}{8}<\/script>, show that <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-20-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-213\" style=\"width: 13.096em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 11.279em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(0.888em, 1011.28em, 2.985em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-214\"><span class=\"msqrt\" id=\"MathJax-Span-215\"><span style=\"display: inline-block; position: relative; width: 11.279em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.938em, 1010.25em, 4.383em, -999.998em); top: -4.005em; left: 0.981em;\"><span class=\"mrow\" id=\"MathJax-Span-216\"><span class=\"mn\" id=\"MathJax-Span-217\" style=\"font-family: MathJax_Main;\">2<\/span><span class=\"mo\" id=\"MathJax-Span-218\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"msqrt\" id=\"MathJax-Span-219\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 8.53em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1007.5em, 4.336em, -999.998em); top: -4.005em; left: 0.981em;\"><span class=\"mrow\" id=\"MathJax-Span-220\"><span class=\"mn\" id=\"MathJax-Span-221\" style=\"font-family: MathJax_Main;\">2<\/span><span class=\"mo\" id=\"MathJax-Span-222\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"msqrt\" id=\"MathJax-Span-223\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 5.781em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1004.94em, 4.243em, -999.998em); top: -4.005em; left: 0.841em;\"><span class=\"mrow\" id=\"MathJax-Span-224\"><span class=\"mn\" id=\"MathJax-Span-225\" style=\"font-family: MathJax_Main;\">2<\/span><span class=\"mo\" id=\"MathJax-Span-226\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-227\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-228\" style=\"font-family: MathJax_Main; padding-left: 0.189em;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-229\"><\/span><span class=\"mn\" id=\"MathJax-Span-230\" style=\"font-family: MathJax_Main; padding-left: 0.189em;\">4<\/span><span class=\"mi\" id=\"MathJax-Span-231\" style=\"font-family: MathJax_Math-italic;\">\u03b8<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.59em, 1004.89em, 3.917em, -999.998em); top: -4.564em; left: 0.841em;\"><span style=\"display: inline-block; position: relative; width: 4.895em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -4.005em; left: -0.091em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -4.005em; left: 4.196em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 0.422em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 0.981em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 1.54em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 2.099em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 2.612em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 3.171em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 3.73em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.078em, 1000.84em, 4.336em, -999.998em); top: -4.052em; left: 0em;\"><span style=\"font-family: MathJax_Main;\">\u221a<\/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(3.59em, 1007.46em, 3.917em, -999.998em); top: -4.658em; left: 0.981em;\"><span style=\"display: inline-block; position: relative; width: 7.458em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -4.005em; left: -0.091em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -4.005em; left: 6.759em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 0.422em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 0.981em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 1.493em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 2.053em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 2.565em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 3.124em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 3.637em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 4.15em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 4.709em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 5.221em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 5.781em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 6.293em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1001.03em, 4.522em, -999.998em); top: -4.098em; left: 0em;\"><span style=\"font-family: MathJax_Size1;\">\u221a<\/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(3.59em, 1010.21em, 3.917em, -999.998em); top: -4.984em; left: 0.981em;\"><span style=\"display: inline-block; position: relative; width: 10.207em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -4.005em; left: -0.091em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -4.005em; left: 9.508em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 0.422em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 0.981em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 1.493em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 2.053em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 2.565em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 3.124em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 3.684em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 4.196em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 4.755em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 5.268em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 5.827em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 6.34em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 6.899em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 7.458em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 7.971em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 8.53em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 9.042em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.705em, 1001.03em, 4.802em, -999.998em); top: -4.098em; left: 0em;\"><span style=\"font-family: MathJax_Size2;\">\u221a<\/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: -0.7em; border-left: 0px solid; width: 0px; height: 2.219em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-20\">\\sqrt{2+\\sqrt{2+\\sqrt{2+2 \\cos 4 \\theta}}}<\/script> = 2cos(\u03b8\/2)<br> Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7821\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-7.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 7\" width=\"316\" height=\"310\" 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(d) 7\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d)\"><\/p>\n\n\n\n<h3>Question 5.<br> Find the extreme values of<br> (i) cos 2x + cos2x<\/h3>\n\n<p>Answer:<br> cos 2x + cos2x = 2 cos<sup>2<\/sup> x \u2013 1 + cos<sup>2<\/sup> x<br> = 3 cos<sup>2<\/sup> x \u2013 1<br> and 0 \u2264 cos<sup>2<\/sup> x \u2264 1<br> \u21d2 0 \u2264 3 cos<sup>2<\/sup> x \u2264 3<br> \u21d2 -1 \u2264 3 cos<sup>2<\/sup> x \u2013 1 \u2264 2<br> Maximum value = 2<br> and minimum value = -1<br> (or) cos 2x + cos<sup>2<\/sup> x = cos 2x + <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-21-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-232\" style=\"width: 5.082em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.383em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.866em, 1004.2em, 3.963em, -999.998em); top: -3.166em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-233\"><span class=\"mrow\" id=\"MathJax-Span-234\"><span class=\"mo\" id=\"MathJax-Span-235\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-236\"><span style=\"display: inline-block; position: relative; width: 2.891em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1002.75em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -1.396em;\"><span class=\"mrow\" id=\"MathJax-Span-237\"><span class=\"mn\" id=\"MathJax-Span-238\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-239\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-240\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-241\" style=\"font-size: 70.7%;\"><\/span><span class=\"mn\" id=\"MathJax-Span-242\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-243\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">x<\/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-244\" 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, 1002.89em, 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: 2.891em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-245\" 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-21\">\\left(\\frac{1+\\cos 2 x}{2}\\right)<\/script><br> We have -1 &lt; cos 2x \u2264 1<br> \u21d2 -3 \u2264 3 cos 2x \u2264 3<br> \u21d2 -2 \u2264 3 cos 2x + 1 \u2264 4<br> \u21d2 -1 \u2264 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-22-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-246\" style=\"width: 4.289em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.684em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1003.68em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-247\"><span class=\"mfrac\" id=\"MathJax-Span-248\"><span style=\"display: inline-block; position: relative; width: 3.404em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1003.26em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -1.629em;\"><span class=\"mrow\" id=\"MathJax-Span-249\"><span class=\"mn\" id=\"MathJax-Span-250\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-251\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-252\" style=\"font-size: 70.7%;\"><\/span><span class=\"mn\" id=\"MathJax-Span-253\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-254\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">x<\/span><span class=\"mo\" id=\"MathJax-Span-255\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-256\" 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(3.404em, 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, 1003.4em, 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: 3.404em; 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: 1.624em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-22\">\\frac{3 \\cos 2 x+1}{2}<\/script> \u2264 2<br> Maximum value = 2<br> Minimum value = -1<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7820\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-8.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 8\" width=\"325\" height=\"189\" sizes=\"auto, (max-width: 325px) 100vw, 325px\" data-pin-nopin=\"true\"><\/p>\n\n<p>(ii) 3sin<sup>2<\/sup>x + 5 cos<sup>2<\/sup>x<br> Answer:<br> 3sin<sup>2<\/sup>x + 5 cos<sup>2<\/sup>x<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7819\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-9.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 9\" width=\"319\" height=\"192\" sizes=\"auto, (max-width: 319px) 100vw, 319px\" data-pin-nopin=\"true\"><\/p>\n\n\n\n<h3>Question 7.<br> Find the periods for the following functions<br> (i) cos<sup>4<\/sup>x<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7818\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-10.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 10\" width=\"317\" height=\"506\" sizes=\"auto, (max-width: 317px) 100vw, 317px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 10\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d)\"><\/p>\n\n<p>(ii) 2sin<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-23-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-258\" style=\"width: 2.519em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.146em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.447em, 1002.01em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-259\"><span class=\"mrow\" id=\"MathJax-Span-260\"><span class=\"mo\" id=\"MathJax-Span-261\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-262\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.84em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.417em;\"><span class=\"mrow\" id=\"MathJax-Span-263\"><span class=\"mi\" id=\"MathJax-Span-264\" 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 class=\"texatom\" id=\"MathJax-Span-265\"><span class=\"mrow\" id=\"MathJax-Span-266\"><span class=\"mi\" id=\"MathJax-Span-267\" style=\"font-size: 70.7%; font-family: MathJax_Main-bold;\">x<\/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=\"texatom\" id=\"MathJax-Span-268\"><span class=\"mrow\" id=\"MathJax-Span-269\"><span class=\"mn\" id=\"MathJax-Span-270\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.93em, 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: 0.934em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-271\" 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: 1.516em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-23\">\\left(\\frac{\\pi \\mathbf{x}}{\\mathrm{4}}\\right)<\/script> + 3cos<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-24-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-272\" style=\"width: 2.519em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.146em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.447em, 1002.01em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-273\"><span class=\"mrow\" id=\"MathJax-Span-274\"><span class=\"mo\" id=\"MathJax-Span-275\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-276\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.79em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.417em;\"><span class=\"mrow\" id=\"MathJax-Span-277\"><span class=\"mi\" id=\"MathJax-Span-278\" 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 class=\"mi\" id=\"MathJax-Span-279\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">x<\/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-280\" 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, 1000.93em, 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: 0.934em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-281\" 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: 1.516em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-24\">\\left(\\frac{\\pi x}{3}\\right)<\/script><br> Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7817\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-11.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 11\" width=\"256\" height=\"233\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 11\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d)\"><\/p>\n\n<p>(iii) sin<sup>2<\/sup>x + 2 cos<sup>2<\/sup>x<br> Answer:<br> Let f(x) = sin<sup>2<\/sup>x + 2cos<sup>2<\/sup>x<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7816\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-12.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 12\" width=\"269\" height=\"234\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 12\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d)\"><\/p>\n\n\n\n<p>(iv) 2sin(<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-25-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-282\" 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-283\"><span class=\"mfrac\" id=\"MathJax-Span-284\"><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-285\" 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-286\" 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-25\">\\frac{\\pi}{4}<\/script> + x)cos x<br> Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7815 size-full\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-13.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 13\" width=\"317\" height=\"318\" sizes=\"auto, (max-width: 317px) 100vw, 317px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 13\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d)\"><br> Period of f(x) is LCM of [\u03c0, \u03c0] = \u03c0<\/p>\n\n<p>(v) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-26-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-287\" style=\"width: 6.2em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.315em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1005.32em, 2.845em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-288\"><span class=\"mfrac\" id=\"MathJax-Span-289\"><span style=\"display: inline-block; position: relative; width: 5.035em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1004.52em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -2.281em;\"><span class=\"mrow\" id=\"MathJax-Span-290\"><span class=\"mn\" id=\"MathJax-Span-291\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span class=\"mi\" id=\"MathJax-Span-292\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-293\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-294\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">x<\/span><span class=\"mo\" id=\"MathJax-Span-295\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-296\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-297\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-298\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-299\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">x<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1004.85em, 4.196em, -999.998em); top: -3.632em; left: 50%; margin-left: -2.467em;\"><span class=\"mrow\" id=\"MathJax-Span-300\"><span class=\"mn\" id=\"MathJax-Span-301\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span class=\"mi\" id=\"MathJax-Span-302\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-303\" style=\"font-size: 70.7%;\"><\/span><span class=\"mn\" id=\"MathJax-Span-304\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-305\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">x<\/span><span class=\"mo\" id=\"MathJax-Span-306\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-307\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span class=\"mi\" id=\"MathJax-Span-308\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-309\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-310\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">x<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1005.03em, 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: 5.035em; 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: 1.678em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-26\">\\frac{5 \\sin x+3 \\cos x}{4 \\sin 2 x+5 \\cos x}<\/script><br> Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7814\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-14.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 14\" width=\"320\" height=\"185\" sizes=\"auto, (max-width: 320px) 100vw, 320px\" data-pin-nopin=\"true\"><\/p>\n\n<h3>II.<br> Question 1.<br> (i) If 0 &lt; A &lt; <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-27-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-311\" style=\"width: 2.006em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.726em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.447em, 1001.59em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-312\"><span class=\"mrow\" id=\"MathJax-Span-313\"><span class=\"mo\" id=\"MathJax-Span-314\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-315\"><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-316\" 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-317\" 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 class=\"mo\" id=\"MathJax-Span-318\" 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: 1.516em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-27\">\\left(\\frac{\\pi}{4}\\right)<\/script>, and cos A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-28-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-319\" 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-320\"><span class=\"mfrac\" id=\"MathJax-Span-321\"><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-322\" 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-323\" 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-28\">\\frac{4}{5}<\/script>, find the values of sin 2A and cos 2A.<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7813\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-15.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 15\" width=\"315\" height=\"235\" sizes=\"auto, (max-width: 315px) 100vw, 315px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 15\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d)\"><\/p>\n\n\n\n<p>(ii) For what values of A in the first quadrant, the expression <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-29-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-324\" style=\"width: 5.874em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.035em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1005.03em, 2.938em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-325\"><span class=\"mfrac\" id=\"MathJax-Span-326\"><span style=\"display: inline-block; position: relative; width: 4.755em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.264em, 1004.62em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -2.328em;\"><span class=\"mrow\" id=\"MathJax-Span-327\"><span class=\"msubsup\" id=\"MathJax-Span-328\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.89em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-329\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cot<\/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-330\" style=\"font-size: 50%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-331\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-332\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-333\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-334\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-335\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">cot<\/span><span class=\"mo\" id=\"MathJax-Span-336\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-337\" 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(3.311em, 1003.36em, 4.196em, -999.998em); top: -3.539em; left: 50%; margin-left: -1.675em;\"><span class=\"mrow\" id=\"MathJax-Span-338\"><span class=\"mn\" id=\"MathJax-Span-339\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"msubsup\" id=\"MathJax-Span-340\" style=\"padding-left: 0.189em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.89em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-341\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cot<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.238em; left: 0.934em;\"><span class=\"mn\" id=\"MathJax-Span-342\" 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-343\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-344\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-345\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-346\" 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(0.888em, 1004.75em, 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: 4.755em; 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: -0.646em; border-left: 0px solid; width: 0px; height: 1.949em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-29\">\\frac{\\cot ^3 A-3 \\cot A}{3 \\cot ^2 A-1}<\/script> is positive?<br> Answer:<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-30-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-347\" style=\"width: 5.874em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.035em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1005.03em, 2.938em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-348\"><span class=\"mfrac\" id=\"MathJax-Span-349\"><span style=\"display: inline-block; position: relative; width: 4.755em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.264em, 1004.62em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -2.328em;\"><span class=\"mrow\" id=\"MathJax-Span-350\"><span class=\"msubsup\" id=\"MathJax-Span-351\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.89em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-352\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cot<\/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-353\" style=\"font-size: 50%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-354\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-355\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-356\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-357\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-358\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">cot<\/span><span class=\"mo\" id=\"MathJax-Span-359\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-360\" 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(3.311em, 1003.36em, 4.196em, -999.998em); top: -3.539em; left: 50%; margin-left: -1.675em;\"><span class=\"mrow\" id=\"MathJax-Span-361\"><span class=\"mn\" id=\"MathJax-Span-362\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"msubsup\" id=\"MathJax-Span-363\" style=\"padding-left: 0.189em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.89em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-364\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cot<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.238em; left: 0.934em;\"><span class=\"mn\" id=\"MathJax-Span-365\" 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-366\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-367\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-368\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-369\" 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(0.888em, 1004.75em, 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: 4.755em; 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: -0.646em; border-left: 0px solid; width: 0px; height: 1.949em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-30\">\\frac{\\cot ^3 A-3 \\cot A}{3 \\cot ^2 A-1}<\/script> &gt; 0<br> \u21d2 cot<sup>3<\/sup>A \u2013 3 cot A &gt; 0<br> \u21d2 cot A (cot<sup>2<\/sup> A \u2013 3) &gt; 0<br> \u21d2 cot A &gt; 0 or cot<sup>2<\/sup> A &gt; 3<br> \u21d2 cot A &gt; \u221a3<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7812\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-16.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 16\" width=\"247\" height=\"270\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 16\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d)\"><\/p>\n\n\n\n<p>(iii) Prove that <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-31-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-370\" style=\"width: 5.641em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.849em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.167em, 1004.85em, 2.891em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-371\"><span class=\"mfrac\" id=\"MathJax-Span-372\"><span style=\"display: inline-block; position: relative; width: 4.569em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1004.43em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -2.234em;\"><span class=\"mrow\" id=\"MathJax-Span-373\"><span class=\"mi\" id=\"MathJax-Span-374\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-375\" style=\"font-size: 70.7%;\"><\/span><span class=\"mn\" id=\"MathJax-Span-376\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-377\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-378\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-379\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-380\" style=\"font-size: 70.7%;\"><\/span><span class=\"mn\" id=\"MathJax-Span-381\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-382\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1003.73em, 4.196em, -999.998em); top: -3.586em; left: 50%; margin-left: -1.862em;\"><span class=\"mrow\" id=\"MathJax-Span-383\"><span class=\"mi\" id=\"MathJax-Span-384\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-385\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-386\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-387\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-388\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-389\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-390\" 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, 1004.57em, 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: 4.569em; 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: 1.786em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-31\">\\frac{\\cos 3 A+\\sin 3 A}{\\cos A-\\sin A}<\/script> = 1 + 2 sin 2A<br> Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7811\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-17.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 17\" width=\"320\" height=\"256\" 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(d) 17\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d)\"><\/p>\n\n<h3>Question 2.<br> (i) Prove that cot(<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-32-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-391\" 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-392\"><span class=\"mfrac\" id=\"MathJax-Span-393\"><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-394\" 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-395\" 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-32\">\\frac{\\pi}{4}<\/script> \u2013 \u03b8) = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-33-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-396\" style=\"width: 3.544em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.031em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1003.03em, 2.891em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-397\"><span class=\"mfrac\" id=\"MathJax-Span-398\"><span style=\"display: inline-block; position: relative; width: 2.752em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1001.77em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.883em;\"><span class=\"mrow\" id=\"MathJax-Span-399\"><span class=\"mi\" id=\"MathJax-Span-400\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-401\" style=\"font-size: 70.7%;\"><\/span><span class=\"mn\" id=\"MathJax-Span-402\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-403\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03b8<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1002.61em, 4.196em, -999.998em); top: -3.632em; left: 50%; margin-left: -1.302em;\"><span class=\"mrow\" id=\"MathJax-Span-404\"><span class=\"mn\" id=\"MathJax-Span-405\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-406\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-407\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-408\" style=\"font-size: 70.7%;\"><\/span><span class=\"mn\" id=\"MathJax-Span-409\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-410\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03b8<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1002.75em, 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: 2.752em; 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: 1.678em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-33\">\\frac{\\cos 2 \\theta}{1-\\sin 2 \\theta}<\/script> and hence find the value of cot 15\u00b0.<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7810\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-18.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 18\" width=\"298\" height=\"539\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 18\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d)\"><\/p>\n\n\n\n<p>(ii) If \u03b8 lies in third quadrant and sin \u03b8 = \u2013<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-34-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-411\" 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-412\"><span class=\"mfrac\" id=\"MathJax-Span-413\"><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-414\" 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-415\" 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-34\">\\frac{4}{5}<\/script>, find the values of cosec<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-35-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-416\" style=\"width: 1.959em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.68em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.4em, 1001.54em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-417\"><span class=\"mrow\" id=\"MathJax-Span-418\"><span class=\"mo\" id=\"MathJax-Span-419\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-420\"><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(3.357em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-421\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03b8<\/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-422\" 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-423\" 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-35\">\\left(\\frac{\\theta}{2}\\right)<\/script> and tan<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-36-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-424\" style=\"width: 1.959em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.68em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.4em, 1001.54em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-425\"><span class=\"mrow\" id=\"MathJax-Span-426\"><span class=\"mo\" id=\"MathJax-Span-427\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-428\"><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(3.357em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-429\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03b8<\/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-430\" 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-431\" 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-36\">\\left(\\frac{\\theta}{2}\\right)<\/script>.<br> Answer:<br> Given \u03b8 lies in third quadrant.<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7809\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-19.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 19\" width=\"316\" height=\"298\" 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(d) 19\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d)\"><\/p>\n\n\n\n<p>(iii) If 450\u00b0 &lt; \u03b8 &lt; 540\u00b0 and sin \u03b8 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-37-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-432\" 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-433\"><span class=\"mfrac\" id=\"MathJax-Span-434\"><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, 1000.65em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-435\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">12<\/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-436\" 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-37\">\\frac{12}{13}<\/script>, then calculate sin<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-38-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-437\" style=\"width: 1.959em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.68em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.4em, 1001.54em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-438\"><span class=\"mrow\" id=\"MathJax-Span-439\"><span class=\"mo\" id=\"MathJax-Span-440\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-441\"><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(3.357em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-442\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03b8<\/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-443\" 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-444\" 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-38\">\\left(\\frac{\\theta}{2}\\right)<\/script> and cos<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-39-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-445\" style=\"width: 1.959em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.68em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.4em, 1001.54em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-446\"><span class=\"mrow\" id=\"MathJax-Span-447\"><span class=\"mo\" id=\"MathJax-Span-448\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-449\"><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(3.357em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-450\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03b8<\/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-451\" 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-452\" 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-39\">\\left(\\frac{\\theta}{2}\\right)<\/script>.<br> Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7808\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-20.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 20\" width=\"320\" height=\"451\" 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(d) 20\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d)\"><\/p>\n\n\n\n<p>(iv) Prove that <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-40-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-453\" style=\"width: 12.491em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 10.767em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.307em, 1010.77em, 3.078em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-454\"><span class=\"mfrac\" id=\"MathJax-Span-455\"><span style=\"display: inline-block; position: relative; width: 2.612em; 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-456\" 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, 1002.52em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -1.256em;\"><span class=\"mrow\" id=\"MathJax-Span-457\"><span class=\"mi\" id=\"MathJax-Span-458\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-459\" style=\"font-size: 70.7%;\"><\/span><span class=\"msubsup\" id=\"MathJax-Span-460\" style=\"padding-left: 0.189em;\"><span style=\"display: inline-block; position: relative; width: 1.354em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.03em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-461\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">290<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.285em; left: 1.074em;\"><span class=\"texatom\" id=\"MathJax-Span-462\"><span class=\"mrow\" id=\"MathJax-Span-463\"><span class=\"mo\" id=\"MathJax-Span-464\" 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, 1002.61em, 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: 2.612em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-465\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mfrac\" id=\"MathJax-Span-466\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 3.637em; 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-467\" 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(3.311em, 1003.54em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -1.768em;\"><span class=\"mrow\" id=\"MathJax-Span-468\"><span class=\"msqrt\" id=\"MathJax-Span-469\"><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-470\"><span class=\"mn\" id=\"MathJax-Span-471\" 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=\"mi\" id=\"MathJax-Span-472\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-473\" style=\"font-size: 70.7%;\"><\/span><span class=\"msubsup\" id=\"MathJax-Span-474\" style=\"padding-left: 0.189em;\"><span style=\"display: inline-block; position: relative; width: 1.354em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.03em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-475\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">250<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.285em; left: 1.074em;\"><span class=\"texatom\" id=\"MathJax-Span-476\"><span class=\"mrow\" id=\"MathJax-Span-477\"><span class=\"mo\" id=\"MathJax-Span-478\" 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, 1003.64em, 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: 3.637em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-479\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-480\" style=\"padding-left: 0.282em;\"><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(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-481\" 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(3.311em, 1000.93em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -0.464em;\"><span class=\"msqrt\" id=\"MathJax-Span-482\"><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-483\"><span class=\"mn\" id=\"MathJax-Span-484\" 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(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><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: 1.841em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-40\">\\frac{1}{\\cos 290^{\\circ}}+\\frac{1}{\\sqrt{3} \\sin 250^{\\circ}}=\\frac{4}{\\sqrt{3}}<\/script><br> Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7807\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-21.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 21\" width=\"317\" height=\"523\" sizes=\"auto, (max-width: 317px) 100vw, 317px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d) 21\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(d)\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>I. Question 1. Simplify (i) sin2\u03b81+cos2\u03b8 Answer: (ii) 3cos\u03b8+cos3\u03b83sin\u03b8\u2212sin3\u03b8 Answer: Question 2. Evaluate the following (i) 6sin 20\u00b0 \u2013 8 sin3 20\u00b0 Answer: 2(3 sin 20\u00b0 \u2013 4sin3 20\u00b0) (Formula) = 2.sin (3 \u00d7 20\u00b0) = 2 sin 60\u00b0 = 23\u221a\u22122 = \u221a3 (ii) cos272\u00b0 \u2013 sin254\u00b0 Answer: cos272\u00b0 \u2013 sin254\u00b0 = sin218\u00b0 \u2013 cos236\u00b0 [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1],"tags":[],"class_list":{"0":"post-3369","1":"post","2":"type-post","3":"status-publish","4":"format-standard","6":"category-other"},"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(d)<\/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-6d\/\" \/>\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(d)\" \/>\n<meta property=\"og:description\" content=\"I. Question 1. Simplify (i) sin2\u03b81+cos2\u03b8 Answer: (ii) 3cos\u03b8+cos3\u03b83sin\u03b8\u2212sin3\u03b8 Answer: Question 2. Evaluate the following (i) 6sin 20\u00b0 \u2013 8 sin3 20\u00b0 Answer: 2(3 sin 20\u00b0 \u2013 4sin3 20\u00b0) (Formula) = 2.sin (3 \u00d7 20\u00b0) = 2 sin 60\u00b0 = 23\u221a\u22122 = \u221a3 (ii) cos272\u00b0 \u2013 sin254\u00b0 Answer: cos272\u00b0 \u2013 sin254\u00b0 = sin218\u00b0 \u2013 cos236\u00b0 [&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-6d\/\" \/>\n<meta property=\"og:site_name\" content=\"Manabadi Boards\" \/>\n<meta property=\"article:published_time\" content=\"2026-01-31T06:39:37+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2026-02-02T07:30:18+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6d-1.png\" \/>\n<meta name=\"author\" content=\"Junaid\" \/>\n<meta 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Question 1. Simplify (i) sin2\u03b81+cos2\u03b8 Answer: (ii) 3cos\u03b8+cos3\u03b83sin\u03b8\u2212sin3\u03b8 Answer: Question 2. 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