{"id":3066,"date":"2026-01-30T14:33:10","date_gmt":"2026-01-30T09:03:10","guid":{"rendered":"https:\/\/www.manabadi.co.in\/boards\/?p=3066"},"modified":"2026-01-30T14:33:14","modified_gmt":"2026-01-30T09:03:14","slug":"ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6a","status":"publish","type":"post","link":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6a\/","title":{"rendered":"TS Inter 1st Year Maths 1A Trigonometric Ratios upto Transformations Solutions Exercise 6(a)"},"content":{"rendered":"\n<h3>I.<br> Question 1.<br> Convert the following into simplest form<br> (i) tan (\u03b8 \u2013 14\u03c0)<\/h3>\n\n<p>Answer:<br> tan (\u03b8 \u2013 14\u03c0) = tan [- (14\u03c0 \u2013 \u03b8)]<br> = \u2013 tan (14\u03c0 \u2013 \u03b8)<br> = \u2013 tan [ 2(7\u03c0) \u2013 \u03b8)<br> = \u2013 tan (-\u03b8) = tan \u03b8<\/p>\n\n<p>(ii) cot (<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: 1.726em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.49em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2\"><span class=\"mfrac\" id=\"MathJax-Span-3\"><span style=\"display: inline-block; position: relative; width: 1.214em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.12em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.557em;\"><span class=\"mrow\" id=\"MathJax-Span-4\"><span class=\"mn\" id=\"MathJax-Span-5\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">21<\/span><span class=\"mi\" id=\"MathJax-Span-6\" 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-7\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.21em, 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: 1.214em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-1\">\\frac{21 \\pi}{2}<\/script> \u2013 \u03b8)<br> Answer:<br> cot (<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-2-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-8\" style=\"width: 1.726em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.49em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-9\"><span class=\"mfrac\" id=\"MathJax-Span-10\"><span style=\"display: inline-block; position: relative; width: 1.214em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.12em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.557em;\"><span class=\"mrow\" id=\"MathJax-Span-11\"><span class=\"mn\" id=\"MathJax-Span-12\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">21<\/span><span class=\"mi\" id=\"MathJax-Span-13\" 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-14\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.21em, 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: 1.214em; 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-2\">\\frac{21 \\pi}{2}<\/script> \u2013 \u03b8) = cot[10\u03c0 + (<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-3-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-15\" 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-16\"><span class=\"mfrac\" id=\"MathJax-Span-17\"><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-18\" 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-19\" 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-3\">\\frac{\\pi}{2}<\/script> \u2013 \u03b8)]<br> = cot (<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-4-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-20\" 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-21\"><span class=\"mfrac\" id=\"MathJax-Span-22\"><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-23\" 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-24\" 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-4\">\\frac{\\pi}{2}<\/script> \u2013 \u03b8) = tan \u03b8<\/p>\n\n<p>(iii) cosec (5\u03c0 + \u03b8)<br> Answer:<br> cosec (5\u03c0 + \u03b8) = cosec [4\u03c0 + (\u03c0 + \u03b8)]<br> = cosec(\u03c0 + \u03b8) = \u2013 cosec \u03b8<\/p>\n\n<p>(iv) sec (4\u03c0 \u2013 \u03b8)<br> Answer:<br> sec (4\u03c0 \u2013 \u03b8) = sec [2(2\u03c0) \u2013 \u03b8]<br> = sec (- \u03b8) = sec \u03b8<\/p>\n\n<h3>Question 2.<br> Find the values of each of the following<br> (i) sin (-405\u00b0)<\/h3>\n\n<p>Answer:<br> sin (-405\u00b0) = -sin 405\u00b0 = -sin (360\u00b0+45\u00b0)<br> = \u2013 sin 45\u00b0 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-5-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-25\" 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(1.307em, 1002.15em, 3.078em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-26\"><span class=\"mo\" id=\"MathJax-Span-27\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-28\"><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, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-29\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1000.93em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -0.464em;\"><span class=\"msqrt\" id=\"MathJax-Span-30\"><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-31\"><span class=\"mn\" id=\"MathJax-Span-32\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span 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-5\">-\\frac{1}{\\sqrt{2}}<\/script><\/p>\n\n\n\n<p>(ii) cos <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-6-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-33\" style=\"width: 3.357em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.891em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.4em, 1002.75em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-34\"><span class=\"mrow\" id=\"MathJax-Span-35\"><span class=\"mo\" id=\"MathJax-Span-36\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mo\" id=\"MathJax-Span-37\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-38\"><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-39\"><span class=\"mn\" id=\"MathJax-Span-40\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">7<\/span><span class=\"mi\" id=\"MathJax-Span-41\" 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-42\" 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 class=\"mo\" id=\"MathJax-Span-43\" 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-6\">\\left(-\\frac{7 \\pi}{2}\\right)<\/script><br> Answer:<br> cos <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-7-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-44\" style=\"width: 3.357em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.891em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.4em, 1002.75em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-45\"><span class=\"mrow\" id=\"MathJax-Span-46\"><span class=\"mo\" id=\"MathJax-Span-47\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mo\" id=\"MathJax-Span-48\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-49\"><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-50\"><span class=\"mn\" id=\"MathJax-Span-51\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">7<\/span><span class=\"mi\" id=\"MathJax-Span-52\" 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-53\" 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 class=\"mo\" id=\"MathJax-Span-54\" 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-7\">\\left(-\\frac{7 \\pi}{2}\\right)<\/script> = cos <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-8-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-55\" 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-56\"><span class=\"mfrac\" id=\"MathJax-Span-57\"><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-58\"><span class=\"mn\" id=\"MathJax-Span-59\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">7<\/span><span class=\"mi\" id=\"MathJax-Span-60\" 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-61\" 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-8\">\\frac{7 \\pi}{2}<\/script> = cos (630\u00b0)<br> = cos (360 + 270\u00b0) = cos 270\u00b0<br> = cos (180 + 90) = -cos 90 = 0<br> (or) cos <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-9-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-62\" style=\"width: 3.357em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.891em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.4em, 1002.75em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-63\"><span class=\"mrow\" id=\"MathJax-Span-64\"><span class=\"mo\" id=\"MathJax-Span-65\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mo\" id=\"MathJax-Span-66\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-67\"><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-68\"><span class=\"mn\" id=\"MathJax-Span-69\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">7<\/span><span class=\"mi\" id=\"MathJax-Span-70\" 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-71\" 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 class=\"mo\" id=\"MathJax-Span-72\" 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-9\">\\left(-\\frac{7 \\pi}{2}\\right)<\/script> = 0 (\u2235 cos(2n + 1)<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-10-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-73\" 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-74\"><span class=\"mfrac\" id=\"MathJax-Span-75\"><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-76\" 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-77\" 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-10\">\\frac{\\pi}{2}<\/script> = 0)<\/p>\n\n<p>(iii) sec (2100\u00b0)<br> Sol. sec (2100\u00b0) = sec [5 \u00d7 360\u00b0 + 300\u00b0]<br> = sec 300\u00b0 = sec (360\u00b0 \u2013 60\u00b0)<br> = sec 60\u00b0 = 2<\/p>\n\n<p>(iv) cot (-315\u00b0)<br> Answer:<br> cot (-315\u00b0) = \u2013 cot 315\u00b0 = \u2013 cot (270 + 45\u00b0)<br> = cot 45\u00b0 = 1<\/p>\n\n\n\n<h3>Question 3.<br> Evaluate<br> (i) cos<sup>2<\/sup> 45\u00b0 + cos<sup>2<\/sup> 135\u00b0 + cos<sup>2<\/sup> 225\u00b0 + cos<sup>2<\/sup> 315\u00b0<\/h3>\n\n<p>Answer:<br> cos 45\u00b0 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-11-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-78\" style=\"width: 1.587em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.354em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.35em, 3.031em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-79\"><span class=\"mfrac\" id=\"MathJax-Span-80\"><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, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-81\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1000.93em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -0.464em;\"><span class=\"msqrt\" id=\"MathJax-Span-82\"><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-83\"><span class=\"mn\" id=\"MathJax-Span-84\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span 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: 2.286em;\"><\/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-11\">\\frac{1}{\\sqrt{2}}<\/script>, cos 135\u00b0 = cos (180 \u2013 45\u00b0)<br> = \u2013 cos 45\u00b0 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-12-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-85\" 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(1.307em, 1002.15em, 3.078em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-86\"><span class=\"mo\" id=\"MathJax-Span-87\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-88\"><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, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-89\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1000.93em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -0.464em;\"><span class=\"msqrt\" id=\"MathJax-Span-90\"><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-91\"><span class=\"mn\" id=\"MathJax-Span-92\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span 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-12\">-\\frac{1}{\\sqrt{2}}<\/script><\/p>\n\n<p>cos 225\u00b0 = cos (180 + 45\u00b0)<br> = \u2013 cos 45\u00b0 = \u2013<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-13-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-93\" style=\"width: 1.587em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.354em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.35em, 3.031em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-94\"><span class=\"mfrac\" id=\"MathJax-Span-95\"><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, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-96\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1000.93em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -0.464em;\"><span class=\"msqrt\" id=\"MathJax-Span-97\"><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-98\"><span class=\"mn\" id=\"MathJax-Span-99\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span 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: 2.286em;\"><\/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-13\">\\frac{1}{\\sqrt{2}}<\/script><\/p>\n\n\n\n<p>cos 315\u00b0 = cos(360 \u2013 45\u00b0)<br> = cos 45\u00b0 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-14-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-100\" style=\"width: 1.587em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.354em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.35em, 3.031em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-101\"><span class=\"mfrac\" id=\"MathJax-Span-102\"><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, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-103\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1000.93em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -0.464em;\"><span class=\"msqrt\" id=\"MathJax-Span-104\"><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-105\"><span class=\"mn\" id=\"MathJax-Span-106\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span 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: 2.286em;\"><\/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-14\">\\frac{1}{\\sqrt{2}}<\/script><\/p>\n\n<p>\u2234 cos<sup>2<\/sup> 45\u00b0 + cos<sup>2<\/sup> 135\u00b0 + cos<sup>2<\/sup> 225\u00b0 + cos<sup>2<\/sup> 315\u00b0<br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-15-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-107\" style=\"width: 7.831em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.759em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.307em, 1006.76em, 2.845em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-108\"><span class=\"mfrac\" id=\"MathJax-Span-109\"><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-110\" 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-111\" 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-112\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mfrac\" id=\"MathJax-Span-113\" style=\"padding-left: 0.235em;\"><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-114\" 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-115\" 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-116\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mfrac\" id=\"MathJax-Span-117\" style=\"padding-left: 0.235em;\"><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-118\" 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-119\" 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-120\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mfrac\" id=\"MathJax-Span-121\" style=\"padding-left: 0.235em;\"><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-122\" 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-123\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-15\">\\frac{1}{2}+\\frac{1}{2}+\\frac{1}{2}+\\frac{1}{2}<\/script> = 2<\/p>\n\n<h3>(ii) sin<sup>2<\/sup><span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-16-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-124\" 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-125\"><span class=\"mfrac\" id=\"MathJax-Span-126\"><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-127\"><span class=\"mn\" id=\"MathJax-Span-128\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-129\" 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-130\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-16\">\\frac{2 \\pi}{3}<\/script> + cos<sup>2<\/sup><span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-17-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-131\" 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-132\"><span class=\"mfrac\" id=\"MathJax-Span-133\"><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-134\"><span class=\"mn\" id=\"MathJax-Span-135\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span class=\"mi\" id=\"MathJax-Span-136\" 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-137\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">6<\/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{5 \\pi}{6}<\/script> \u2013 tan<sup>2<\/sup><span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-18-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-138\" 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-139\"><span class=\"mfrac\" id=\"MathJax-Span-140\"><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-141\"><span class=\"mn\" id=\"MathJax-Span-142\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-143\" 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-144\" 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, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-18\">\\frac{3 \\pi}{4}<\/script><\/h3>\n\n<p>Answer:<br> <img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-7426\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6a-1.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a) 1\" width=\"319\" height=\"176\" sizes=\"(max-width: 319px) 100vw, 319px\" data-pin-nopin=\"true\"><\/p>\n\n\n\n<p>(iii) cos 225\u00b0 \u2013 sin 225\u00b0 + tan 495\u00b0 \u2013 cot 495\u00b0<br> Answer:<br> cot (180 + 45) \u2013 sIn (180 + 45) + tan (360 + 135) \u2013 cot (360 + 135)<br> = \u2013 cot 45\u00b0 + sin 45\u00b0 + tan 135 \u2013 cot 135\u00b0<br> = \u2013 cos 45\u00b0 + sin 45\u00b0 +tan(180 \u2013 45) \u2013 cot(180 \u2013 45)<br> = \u2013 cos 45\u00b0 + sin 45\u00b0 \u2013 tan 45\u00b0 + cot 45\u00b0<br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-19-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-145\" style=\"width: 5.548em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.755em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.307em, 1004.75em, 3.078em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-146\"><span class=\"mo\" id=\"MathJax-Span-147\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-148\"><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, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-149\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1000.93em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -0.464em;\"><span class=\"msqrt\" id=\"MathJax-Span-150\"><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-151\"><span class=\"mn\" id=\"MathJax-Span-152\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.07em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.074em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-153\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mfrac\" id=\"MathJax-Span-154\" style=\"padding-left: 0.235em;\"><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, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-155\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1000.93em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -0.464em;\"><span class=\"msqrt\" id=\"MathJax-Span-156\"><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-157\"><span class=\"mn\" id=\"MathJax-Span-158\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span 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-19\">-\\frac{1}{\\sqrt{2}}+\\frac{1}{\\sqrt{2}}<\/script> \u2013 1 + 1 = 0<\/p>\n\n<p>(iv) (cos \u03b8 \u2013 sin \u03b8) if<br> (a) \u03b8 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-20-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-159\" 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-160\"><span class=\"mfrac\" id=\"MathJax-Span-161\"><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-162\"><span class=\"mn\" id=\"MathJax-Span-163\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">7<\/span><span class=\"mi\" id=\"MathJax-Span-164\" 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-165\" 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, 1000.89em, 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-20\">\\frac{7 \\pi}{4}<\/script><br> (b) \u03b8 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-21-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-166\" style=\"width: 1.726em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.49em, 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: 1.214em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.12em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.557em;\"><span class=\"mrow\" id=\"MathJax-Span-169\"><span class=\"mn\" id=\"MathJax-Span-170\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">11<\/span><span class=\"mi\" id=\"MathJax-Span-171\" 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-172\" 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, 1001.21em, 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: 1.214em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-21\">\\frac{11 \\pi}{4}<\/script><br> Answer:<br> <img decoding=\"async\" class=\"alignnone size-full wp-image-7425\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6a-2.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a) 2\" width=\"308\" height=\"288\" sizes=\"(max-width: 308px) 100vw, 308px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a) 2\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a)\"><\/p>\n\n<h3>Question 4.<br> (i) If sin \u03b8 = \u2013<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-22-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-173\" 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-174\"><span class=\"mfrac\" id=\"MathJax-Span-175\"><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-176\" 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-177\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-22\">\\frac{1}{3}<\/script> and 0 does not lie in the third 3 quadrant, find the values of (a) cos \u03b8 and (b) cot \u03b8. (March 2013)<\/h3>\n\n\n\n<p>Answer:<br> sin \u03b8 = \u2013<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-23-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-178\" 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-179\"><span class=\"mfrac\" id=\"MathJax-Span-180\"><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-181\" 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-182\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-23\">\\frac{1}{3}<\/script> and sin \u03b8 is negative and does not lie in third quadrant,<br> \u21d2 \u03b8 lies in fourth quadrant. In IVth quadrant cos \u03b8 is positive and cot \u03b8 is negative.<br> a) cos \u03b8 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-24-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-183\" style=\"width: 15.007em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 12.91em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.027em, 1012.91em, 3.124em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-184\"><span class=\"msqrt\" id=\"MathJax-Span-185\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1004.06em, 4.243em, -999.998em); top: -4.005em; left: 0.981em;\"><span class=\"mrow\" id=\"MathJax-Span-186\"><span class=\"mn\" id=\"MathJax-Span-187\" style=\"font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-188\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"msubsup\" id=\"MathJax-Span-189\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 1.68em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-190\" style=\"font-family: MathJax_Main;\">sin<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 1.214em;\"><span class=\"mn\" id=\"MathJax-Span-191\" style=\"font-size: 70.7%; 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-192\"><\/span><span class=\"mi\" id=\"MathJax-Span-193\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b8<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.59em, 1004.01em, 3.917em, -999.998em); top: -4.704em; left: 0.981em;\"><span style=\"display: inline-block; position: relative; width: 4.01em; 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: 3.311em;\">\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: 0.375em;\">\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: 0.888em;\">\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: 1.4em;\">\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: 1.913em;\">\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: 2.379em;\">\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: 2.891em;\">\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: -4.145em; 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 class=\"mo\" id=\"MathJax-Span-194\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"msqrt\" id=\"MathJax-Span-195\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1002.47em, 4.522em, -999.998em); top: -4.005em; left: 0.981em;\"><span class=\"mrow\" id=\"MathJax-Span-196\"><span class=\"mn\" id=\"MathJax-Span-197\" style=\"font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-198\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-199\" style=\"padding-left: 0.235em;\"><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-200\" 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-201\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">9<\/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: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.59em, 1002.42em, 3.917em, -999.998em); top: -4.89em; left: 0.981em;\"><span style=\"display: inline-block; position: relative; width: 2.425em; 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: 1.726em;\">\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: 0.375em;\">\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: 0.841em;\">\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: 1.307em;\">\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: -4.005em; 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 class=\"mo\" id=\"MathJax-Span-202\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-203\" style=\"padding-left: 0.282em;\"><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(3.264em, 1001.31em, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.65em;\"><span class=\"mrow\" id=\"MathJax-Span-204\"><span class=\"mn\" id=\"MathJax-Span-205\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"msqrt\" id=\"MathJax-Span-206\"><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-207\"><span class=\"mn\" id=\"MathJax-Span-208\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-209\" 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, 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: 2.332em;\"><\/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-24\">\\sqrt{1-\\sin ^2 \\theta}=\\sqrt{1-\\frac{1}{9}}=\\frac{2 \\sqrt{2}}{3}<\/script><br> b) cot \u03b8 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-25-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-210\" style=\"width: 2.192em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.866em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.87em, 2.845em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-211\"><span class=\"mfrac\" id=\"MathJax-Span-212\"><span style=\"display: inline-block; position: relative; width: 1.587em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.697em;\"><span class=\"mrow\" id=\"MathJax-Span-213\"><span class=\"mi\" id=\"MathJax-Span-214\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-215\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-216\" 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, 1001.35em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.697em;\"><span class=\"mrow\" id=\"MathJax-Span-217\"><span class=\"mi\" id=\"MathJax-Span-218\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-219\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-220\" 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, 1001.59em, 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: 1.587em; 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-25\">\\frac{\\cos \\theta}{\\sin \\theta}<\/script> = -2\u221a2<\/p>\n\n\n\n<p>(ii) If cos \u03b8 = t (0 &lt; t &lt; 1) and \u03b8 does not lie in the first quadrant, find the values of a) sin \u03b8 b) tan \u03b8<br> Answer:<br> cos \u03b8 = t, (0 &lt; t &lt; 1)<br> \u21d2 cos \u03b8 is positive and 0 does not lie in first quadrant<br> \u21d2 \u03b8 lies in IVth quadrant<br> a) sin \u03b8 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-26-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-221\" style=\"width: 13.05em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 11.233em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.354em, 1011.23em, 2.612em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-222\"><span class=\"mo\" id=\"MathJax-Span-223\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"msqrt\" id=\"MathJax-Span-224\"><span style=\"display: inline-block; position: relative; width: 4.988em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.124em, 1004.15em, 4.243em, -999.998em); top: -4.005em; left: 0.841em;\"><span class=\"mrow\" id=\"MathJax-Span-225\"><span class=\"mn\" id=\"MathJax-Span-226\" style=\"font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-227\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"msubsup\" id=\"MathJax-Span-228\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 1.773em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.31em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-229\" style=\"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: 1.354em;\"><span class=\"mn\" id=\"MathJax-Span-230\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-231\"><\/span><span class=\"mi\" id=\"MathJax-Span-232\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b8<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.59em, 1004.1em, 3.917em, -999.998em); top: -4.564em; left: 0.841em;\"><span style=\"display: inline-block; position: relative; width: 4.103em; 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: 3.404em;\">\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: 0.934em;\">\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: 1.447em;\">\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: 1.959em;\">\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: 2.472em;\">\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: 2.985em;\">\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 class=\"mo\" id=\"MathJax-Span-233\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mo\" id=\"MathJax-Span-234\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">\u2212<\/span><span class=\"msqrt\" id=\"MathJax-Span-235\"><span style=\"display: inline-block; position: relative; width: 3.357em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.124em, 1002.52em, 4.243em, -999.998em); top: -4.005em; left: 0.841em;\"><span class=\"mrow\" id=\"MathJax-Span-236\"><span class=\"mn\" id=\"MathJax-Span-237\" style=\"font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-238\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"msubsup\" id=\"MathJax-Span-239\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.795em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.264em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-240\" style=\"font-family: MathJax_Math-italic;\">t<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.285em; left: 0.375em;\"><span class=\"mn\" id=\"MathJax-Span-241\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/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, 1002.47em, 3.917em, -999.998em); top: -4.564em; left: 0.841em;\"><span style=\"display: inline-block; position: relative; width: 2.472em; 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: 1.773em;\">\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: 0.375em;\">\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: 0.888em;\">\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: 1.354em;\">\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: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.214em; border-left: 0px solid; width: 0px; height: 1.246em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-26\">-\\sqrt{1-\\cos ^2 \\theta}=-\\sqrt{1-t^2}<\/script><br> b) tan \u03b8 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-27-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-242\" style=\"width: 7.644em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.573em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(0.934em, 1006.57em, 2.891em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-243\"><span class=\"mfrac\" id=\"MathJax-Span-244\"><span style=\"display: inline-block; position: relative; width: 1.587em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1001.35em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.697em;\"><span class=\"mrow\" id=\"MathJax-Span-245\"><span class=\"mi\" id=\"MathJax-Span-246\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-247\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-248\" 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, 1001.45em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.697em;\"><span class=\"mrow\" id=\"MathJax-Span-249\"><span class=\"mi\" id=\"MathJax-Span-250\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-251\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-252\" 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, 1001.59em, 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: 1.587em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-253\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mo\" id=\"MathJax-Span-254\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-255\"><span style=\"display: inline-block; position: relative; width: 2.286em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.19em, 4.336em, -999.998em); top: -4.564em; left: 50%; margin-left: -1.069em;\"><span class=\"msqrt\" id=\"MathJax-Span-256\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.264em, 1001.45em, 4.196em, -999.998em); top: -4.005em; left: 0.701em;\"><span class=\"mrow\" id=\"MathJax-Span-257\"><span class=\"mn\" id=\"MathJax-Span-258\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-259\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"msubsup\" id=\"MathJax-Span-260\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.24em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-261\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">t<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.238em; left: 0.235em;\"><span class=\"mn\" id=\"MathJax-Span-262\" 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><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1001.45em, 1.26em, -999.998em); top: -1.768em; left: 0.701em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 1.447em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1000.7em, 4.383em, -999.998em); top: -4.098em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size1;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.24em, 4.15em, -999.998em); top: -3.679em; left: 50%; margin-left: -0.137em;\"><span class=\"mi\" id=\"MathJax-Span-263\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">t<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1002.29em, 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: 2.286em; 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: 2.003em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-27\">\\frac{\\sin \\theta}{\\cos \\theta}=-\\frac{\\sqrt{1-t^2}}{t}<\/script><\/p>\n\n<p>(iii) Find the value of sin 330\u00b0. cos 120\u00b0 + cos 210\u00b0. sin 300\u00b0<br> Answer:<br> sin 330\u00b0 cos 120\u00b0 + cos 210\u00b0 sin 300\u00b0<br> = sin (360 \u2013 30) cos (180 \u2013 60) + cos ( 180 + 30) sin (360 \u2013 60)<br> = (-sin 30\u00b0) (-cos 60\u00b0) + (-cos 30\u00b0) (- sin 60\u00b0)<br> = sin 30 cos 60 + cos 30 sin 60 = sin (30 + 60)<br> = sin 90\u00b0 = 1<\/p>\n\n<p>(iv) If cosec \u03b8 + cot \u03b8 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-28-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-264\" 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-265\"><span class=\"mfrac\" id=\"MathJax-Span-266\"><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-267\" 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-268\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-28\">\\frac{1}{3}<\/script>, find cos \u03b8 and determine the quadrant in which \u03b8 lies.<br>\n\nAnswer:<br> we have coses<sup>2<\/sup>\u03b8 \u2013 cot<sup>2<\/sup> \u03b8 = 1<br> \u21d2 (cosec \u03b8 + cot \u03b8) (cosec \u03b8 \u2013 cot \u03b8) = 1<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7424\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6a-3.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a) 3\" width=\"316\" height=\"445\" 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(a) 3\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a)\"><br> \u2234 sin \u03b8 is positive and cos \u03b8 is negative,<br> \u21d2 \u03b8 lies in IInd quadrant.<\/p>\n\n\n\n<h3>Question 5.<br> (i) If sin \u03b1 + cosec \u03b1= 2, find the value of sin<sup>n<\/sup> \u03b1 + cosec<sup>n<\/sup> \u03b1; n \u2208 Z.<\/h3>\n\n<p>Answer:<br> Given sin \u03b1 + cosec \u03b1 = 2<br> Squaring both sides<br> sin<sup>2<\/sup> \u03b1 = cosec<sup>2<\/sup> \u03b1 + 2 = 4<br> \u21d2 sin \u03b1 + cosec \u03b1 = 2<br> cubing on both sides<br> sin<sup>3<\/sup> \u03b1 + cosec<sup>3<\/sup> \u03b1 + 3 sin \u03b1 cosec \u03b1 (sin \u03b1 + cosec \u03b1) = 8<br> sin<sup>3<\/sup> \u03b1 + cosec<sup>3<\/sup> \u03b1 + 3 (2) = 8<br> \u21d2 sin<sup>3<\/sup> \u03b1 + cosec<sup>3<\/sup> \u03b1 = 2<br> In the same way sin<sup>n<\/sup> \u03b1 + cosec<sup>n<\/sup> \u03b1 = 2 (n \u2208 z)<\/p>\n\n<p>(ii) If sec \u03b8 + tan \u03b8 = 5, find the quadrant in which \u03b8 lies and find the value of sin \u03b8<br> Answer:<br> We have sec<sup>2<\/sup> \u03b8 \u2013 tan<sup>2<\/sup> \u03b8 = 1<br> \u21d2 (sec \u03b8 + tan \u03b8) (sec \u03b8 \u2013 tan \u03b8) = 1<br> Also given sec \u03b8 + tan \u03b8 = 5 \u2026\u2026\u2026\u2026.(2)<\/p>\n\n<p>Adding (1) and (2)<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7423\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6a-4.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a) 4\" width=\"232\" height=\"278\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a) 4\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a)\"><br> tan \u03b8 is +ve, sec \u03b8 is + ve<br> \u21d2 \u03b8 lies is first quadrant.<\/p>\n\n\n\n<h3>II.<br> Question 1.<br> Prove that<br> (i) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-29-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-269\" style=\"width: 12.677em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 10.906em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(0.468em, 1010.91em, 3.59em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-270\"><span class=\"mfrac\" id=\"MathJax-Span-271\"><span style=\"display: inline-block; position: relative; width: 10.627em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.078em, 1009.32em, 4.616em, -999.998em); top: -4.844em; left: 50%; margin-left: -4.704em;\"><span class=\"mrow\" id=\"MathJax-Span-272\"><span class=\"mi\" id=\"MathJax-Span-273\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-274\" style=\"font-size: 70.7%;\"><\/span><span class=\"mo\" id=\"MathJax-Span-275\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">(<\/span><span class=\"mi\" id=\"MathJax-Span-276\" 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=\"mo\" id=\"MathJax-Span-277\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-278\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-279\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">)<\/span><span class=\"mi\" id=\"MathJax-Span-280\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">cot<\/span><span class=\"mo\" id=\"MathJax-Span-281\" style=\"font-size: 70.7%;\"><\/span><span class=\"mrow\" id=\"MathJax-Span-282\"><span class=\"mo\" id=\"MathJax-Span-283\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">(<\/span><\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-284\"><span style=\"display: inline-block; position: relative; width: 0.422em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.637em, 1000.28em, 4.15em, -999.998em); top: -4.331em; left: 50%; margin-left: -0.137em;\"><span class=\"mi\" id=\"MathJax-Span-285\" style=\"font-size: 50%; 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(3.544em, 1000.24em, 4.15em, -999.998em); top: -3.679em; left: 50%; margin-left: -0.137em;\"><span class=\"mn\" id=\"MathJax-Span-286\" style=\"font-size: 50%; 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, 1000.42em, 1.214em, -999.998em); top: -1.209em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.422em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-287\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-288\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-289\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-290\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-291\" style=\"font-size: 70.7%;\"><\/span><span class=\"mo\" id=\"MathJax-Span-292\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">(<\/span><span class=\"mo\" id=\"MathJax-Span-293\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-294\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-295\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">)<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.078em, 1010.44em, 4.616em, -999.998em); top: -3.306em; left: 50%; margin-left: -5.263em;\"><span class=\"mrow\" id=\"MathJax-Span-296\"><span class=\"mi\" id=\"MathJax-Span-297\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">tan<\/span><span class=\"mo\" id=\"MathJax-Span-298\" style=\"font-size: 70.7%;\"><\/span><span class=\"mo\" id=\"MathJax-Span-299\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">(<\/span><span class=\"mi\" id=\"MathJax-Span-300\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-301\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-302\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-303\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">)<\/span><span class=\"mi\" id=\"MathJax-Span-304\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">tan<\/span><span class=\"mo\" id=\"MathJax-Span-305\" style=\"font-size: 70.7%;\"><\/span><span class=\"mrow\" id=\"MathJax-Span-306\"><span class=\"mo\" id=\"MathJax-Span-307\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">(<\/span><\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-308\"><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(3.544em, 1000.56em, 4.15em, -999.998em); top: -4.331em; left: 50%; margin-left: -0.277em;\"><span class=\"mrow\" id=\"MathJax-Span-309\"><span class=\"mn\" id=\"MathJax-Span-310\" style=\"font-size: 50%; font-family: MathJax_Main;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-311\" style=\"font-size: 50%; 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(3.544em, 1000.24em, 4.15em, -999.998em); top: -3.679em; left: 50%; margin-left: -0.137em;\"><span class=\"mn\" id=\"MathJax-Span-312\" style=\"font-size: 50%; 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: -1.209em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.655em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-313\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-314\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-315\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-316\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-317\" style=\"font-size: 70.7%;\"><\/span><span class=\"mo\" id=\"MathJax-Span-318\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">(<\/span><span class=\"mn\" id=\"MathJax-Span-319\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-320\" 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=\"mo\" id=\"MathJax-Span-321\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-322\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-323\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">)<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1010.63em, 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: 10.627em; 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: -1.403em; border-left: 0px solid; width: 0px; height: 3.408em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-29\">\\frac{\\cos (\\pi-A) \\cot \\left(\\frac{\\pi}{2}+A\\right) \\cos (-A)}{\\tan (\\pi+A) \\tan \\left(\\frac{3 \\pi}{2}+A\\right) \\sin (2 \\pi-A)}<\/script> = cos A<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7422\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6a-5.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a) 5\" width=\"312\" height=\"202\"  sizes=\"auto, (max-width: 312px) 100vw, 312px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a) 5\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a)\"><\/p>\n\n\n\n<p>(ii) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-30-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-324\" style=\"width: 14.168em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 12.211em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(0.468em, 1012.21em, 3.59em, -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: 11.932em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.078em, 1010.81em, 4.616em, -999.998em); top: -4.844em; left: 50%; margin-left: -5.45em;\"><span class=\"mrow\" id=\"MathJax-Span-327\"><span class=\"mi\" id=\"MathJax-Span-328\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-329\" style=\"font-size: 70.7%;\"><\/span><span class=\"mo\" id=\"MathJax-Span-330\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">(<\/span><span class=\"mn\" id=\"MathJax-Span-331\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-332\" 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=\"mo\" id=\"MathJax-Span-333\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-334\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-335\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">)<\/span><span class=\"mi\" id=\"MathJax-Span-336\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-337\" style=\"font-size: 70.7%;\"><\/span><span class=\"mrow\" id=\"MathJax-Span-338\"><span class=\"mo\" id=\"MathJax-Span-339\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">(<\/span><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-340\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-341\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-342\"><span style=\"display: inline-block; position: relative; width: 0.422em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.637em, 1000.28em, 4.15em, -999.998em); top: -4.331em; left: 50%; margin-left: -0.137em;\"><span class=\"mi\" id=\"MathJax-Span-343\" style=\"font-size: 50%; 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(3.544em, 1000.24em, 4.15em, -999.998em); top: -3.679em; left: 50%; margin-left: -0.137em;\"><span class=\"mn\" id=\"MathJax-Span-344\" style=\"font-size: 50%; 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, 1000.42em, 1.214em, -999.998em); top: -1.209em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.422em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-345\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-346\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">tan<\/span><span class=\"mo\" id=\"MathJax-Span-347\" style=\"font-size: 70.7%;\"><\/span><span class=\"mrow\" id=\"MathJax-Span-348\"><span class=\"mo\" id=\"MathJax-Span-349\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">(<\/span><\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-350\"><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(3.544em, 1000.56em, 4.15em, -999.998em); top: -4.331em; left: 50%; margin-left: -0.277em;\"><span class=\"mrow\" id=\"MathJax-Span-351\"><span class=\"mn\" id=\"MathJax-Span-352\" style=\"font-size: 50%; font-family: MathJax_Main;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-353\" style=\"font-size: 50%; 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(3.544em, 1000.24em, 4.15em, -999.998em); top: -3.679em; left: 50%; margin-left: -0.137em;\"><span class=\"mn\" id=\"MathJax-Span-354\" style=\"font-size: 50%; 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: -1.209em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.655em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-355\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-356\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-357\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.078em, 1011.7em, 4.616em, -999.998em); top: -3.306em; left: 50%; margin-left: -5.916em;\"><span class=\"mrow\" id=\"MathJax-Span-358\"><span class=\"texatom\" id=\"MathJax-Span-359\"><span class=\"mrow\" id=\"MathJax-Span-360\"><span class=\"mi\" id=\"MathJax-Span-361\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">c<\/span><span class=\"mi\" id=\"MathJax-Span-362\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">o<\/span><span class=\"mi\" id=\"MathJax-Span-363\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">s<\/span><span class=\"mi\" id=\"MathJax-Span-364\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">e<\/span><span class=\"mi\" id=\"MathJax-Span-365\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-366\"><span class=\"mo\" id=\"MathJax-Span-367\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">(<\/span><\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-368\"><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(3.544em, 1000.79em, 4.15em, -999.998em); top: -4.331em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-369\"><span class=\"mn\" id=\"MathJax-Span-370\" style=\"font-size: 50%; font-family: MathJax_Main;\">13<\/span><span class=\"mi\" id=\"MathJax-Span-371\" style=\"font-size: 50%; 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(3.544em, 1000.24em, 4.15em, -999.998em); top: -3.679em; left: 50%; margin-left: -0.137em;\"><span class=\"mn\" id=\"MathJax-Span-372\" style=\"font-size: 50%; 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: -1.209em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-373\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-374\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-375\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-376\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">sec<\/span><span class=\"mo\" id=\"MathJax-Span-377\" style=\"font-size: 70.7%;\"><\/span><span class=\"mo\" id=\"MathJax-Span-378\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">(<\/span><span class=\"mn\" id=\"MathJax-Span-379\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-380\" 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=\"mo\" id=\"MathJax-Span-381\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-382\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-383\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">)<\/span><span class=\"mi\" id=\"MathJax-Span-384\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">cot<\/span><span class=\"mo\" id=\"MathJax-Span-385\" style=\"font-size: 70.7%;\"><\/span><span class=\"mrow\" id=\"MathJax-Span-386\"><span class=\"mo\" id=\"MathJax-Span-387\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">(<\/span><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-388\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><span class=\"mo\" id=\"MathJax-Span-389\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-390\"><span style=\"display: inline-block; position: relative; width: 0.422em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.637em, 1000.28em, 4.15em, -999.998em); top: -4.331em; left: 50%; margin-left: -0.137em;\"><span class=\"mi\" id=\"MathJax-Span-391\" style=\"font-size: 50%; 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(3.544em, 1000.24em, 4.15em, -999.998em); top: -3.679em; left: 50%; margin-left: -0.137em;\"><span class=\"mn\" id=\"MathJax-Span-392\" style=\"font-size: 50%; 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, 1000.42em, 1.214em, -999.998em); top: -1.209em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.422em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-393\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">)<\/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, 1011.93em, 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: 11.932em; 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: -1.403em; border-left: 0px solid; width: 0px; height: 3.408em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-30\">\\frac{\\sin (3 \\pi-A) \\cos \\left(A-\\frac{\\pi}{2}\\right) \\tan \\left(\\frac{3 \\pi}{2}-A\\right)}{{cosec}\\left(\\frac{13 \\pi}{2}+A\\right) \\sec (3 \\pi+A) \\cot \\left(A-\\frac{\\pi}{2}\\right)}<\/script><br> Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7421\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6a-6.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a) 6\" width=\"315\" height=\"249\" 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(a) 6\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a)\"><\/p>\n\n\n\n<p>(iii) sin 780\u00b0. sin 480\u00b0 + cos 240\u00b0. cos 300\u00b0 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-31-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-394\" 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-395\"><span class=\"mfrac\" id=\"MathJax-Span-396\"><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-397\" 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-398\" 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-31\">\\frac{1}{2}<\/script><br> Answer:<br> sin [2 \u00d7 360 + 60] sin [360 + 120] + cos [180 + 60] cos [360-60]<br> = sin 60 sin 120 \u2013 cos 60 cos 60<br> = sin 60 sin 60 \u2013 cos 60. cos 60<br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-32-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-399\" style=\"width: 14.401em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 12.397em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.074em, 1012.4em, 2.845em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-400\"><span class=\"mfrac\" id=\"MathJax-Span-401\"><span style=\"display: inline-block; position: relative; width: 1.074em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.311em, 1000.93em, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.464em;\"><span class=\"msqrt\" id=\"MathJax-Span-402\"><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-403\"><span class=\"mn\" id=\"MathJax-Span-404\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-405\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.07em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.074em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-406\"><span style=\"display: inline-block; position: relative; width: 1.074em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.311em, 1000.93em, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.464em;\"><span class=\"msqrt\" id=\"MathJax-Span-407\"><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-408\"><span class=\"mn\" id=\"MathJax-Span-409\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-410\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.07em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.074em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-411\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-412\" style=\"padding-left: 0.235em;\"><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-413\" 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-414\" 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-415\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u22c5<\/span><span class=\"mfrac\" id=\"MathJax-Span-416\" style=\"padding-left: 0.235em;\"><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-417\" 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-418\" 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-419\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-420\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-421\" 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-422\" 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 class=\"mo\" id=\"MathJax-Span-423\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-424\" style=\"padding-left: 0.235em;\"><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-425\" 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-426\" 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 class=\"mo\" id=\"MathJax-Span-427\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-428\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-429\" 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-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><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.841em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-32\">\\frac{\\sqrt{3}}{2} \\frac{\\sqrt{3}}{2}-\\frac{1}{2} \\cdot \\frac{1}{2}=\\frac{3}{4}-\\frac{1}{4}=\\frac{1}{2}<\/script><\/p>\n\n\n\n<p>(iv) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-33-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-431\" style=\"width: 11.652em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 10.021em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1010.02em, 2.891em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-432\"><span class=\"mfrac\" id=\"MathJax-Span-433\"><span style=\"display: inline-block; position: relative; width: 9.741em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1009.65em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -4.797em;\"><span class=\"mrow\" id=\"MathJax-Span-434\"><span class=\"mi\" id=\"MathJax-Span-435\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-436\" style=\"font-size: 70.7%;\"><\/span><span class=\"msubsup\" id=\"MathJax-Span-437\" 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-438\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">150<\/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-439\"><span class=\"mrow\" id=\"MathJax-Span-440\"><span class=\"mo\" id=\"MathJax-Span-441\" style=\"font-size: 50%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-442\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-443\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span class=\"mi\" id=\"MathJax-Span-444\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-445\" style=\"font-size: 70.7%;\"><\/span><span class=\"msubsup\" id=\"MathJax-Span-446\" 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-447\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">300<\/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-448\"><span class=\"mrow\" id=\"MathJax-Span-449\"><span class=\"mo\" id=\"MathJax-Span-450\" style=\"font-size: 50%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-451\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-452\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">7<\/span><span class=\"mi\" id=\"MathJax-Span-453\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">tan<\/span><span class=\"mo\" id=\"MathJax-Span-454\" style=\"font-size: 70.7%;\"><\/span><span class=\"msubsup\" id=\"MathJax-Span-455\" 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-456\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">225<\/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-457\"><span class=\"mrow\" id=\"MathJax-Span-458\"><span class=\"mo\" id=\"MathJax-Span-459\" 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(3.357em, 1006.06em, 4.196em, -999.998em); top: -3.586em; left: 50%; margin-left: -3.027em;\"><span class=\"mrow\" id=\"MathJax-Span-460\"><span class=\"mi\" id=\"MathJax-Span-461\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">tan<\/span><span class=\"mo\" id=\"MathJax-Span-462\" style=\"font-size: 70.7%;\"><\/span><span class=\"msubsup\" id=\"MathJax-Span-463\" 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-464\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">135<\/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-465\"><span class=\"mrow\" id=\"MathJax-Span-466\"><span class=\"mo\" id=\"MathJax-Span-467\" style=\"font-size: 50%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-468\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-469\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-470\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-471\" style=\"font-size: 70.7%;\"><\/span><span class=\"msubsup\" id=\"MathJax-Span-472\" 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-473\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">210<\/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-474\"><span class=\"mrow\" id=\"MathJax-Span-475\"><span class=\"mo\" id=\"MathJax-Span-476\" 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, 1009.74em, 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: 9.741em; 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-33\">\\frac{\\sin 150^{\\circ}-5 \\cos 300^{\\circ}+7 \\tan 225^{\\circ}}{\\tan 135^{\\circ}+3 \\sin 210^{\\circ}}<\/script> = -2<br> Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7420\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6a-7.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a) 7\" width=\"307\" height=\"175\" sizes=\"auto, (max-width: 307px) 100vw, 307px\" data-pin-nopin=\"true\"><\/p>\n\n\n\n<p>(v) cot<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-34-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-477\" style=\"width: 2.379em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.053em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.447em, 1001.91em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-478\"><span class=\"mrow\" id=\"MathJax-Span-479\"><span class=\"mo\" id=\"MathJax-Span-480\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-481\"><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(3.544em, 1000.42em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-482\" 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, 1000.7em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-483\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">20<\/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 class=\"mo\" id=\"MathJax-Span-484\" 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-34\">\\left(\\frac{\\pi}{20}\\right)<\/script>. cot<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-35-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-485\" 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(1.4em, 1001.96em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-486\"><span class=\"mrow\" id=\"MathJax-Span-487\"><span class=\"mo\" id=\"MathJax-Span-488\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-489\"><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-490\"><span class=\"mn\" id=\"MathJax-Span-491\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-492\" 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, 1000.7em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-493\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">20<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-494\" 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{3 \\pi}{20}\\right)<\/script>. cot<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-36-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-495\" 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(1.4em, 1001.96em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-496\"><span class=\"mrow\" id=\"MathJax-Span-497\"><span class=\"mo\" id=\"MathJax-Span-498\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-499\"><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-500\"><span class=\"mn\" id=\"MathJax-Span-501\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span class=\"mi\" id=\"MathJax-Span-502\" 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, 1000.7em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-503\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">20<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-504\" 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{5 \\pi}{20}\\right)<\/script>. cot<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-37-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-505\" 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(1.4em, 1001.96em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-506\"><span class=\"mrow\" id=\"MathJax-Span-507\"><span class=\"mo\" id=\"MathJax-Span-508\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-509\"><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-510\"><span class=\"mn\" id=\"MathJax-Span-511\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">7<\/span><span class=\"mi\" id=\"MathJax-Span-512\" 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, 1000.7em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-513\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">20<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-514\" 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-37\">\\left(\\frac{7 \\pi}{20}\\right)<\/script>. cot<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-38-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-515\" 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(1.4em, 1001.96em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-516\"><span class=\"mrow\" id=\"MathJax-Span-517\"><span class=\"mo\" id=\"MathJax-Span-518\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-519\"><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-520\"><span class=\"mn\" id=\"MathJax-Span-521\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">9<\/span><span class=\"mi\" id=\"MathJax-Span-522\" 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, 1000.7em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-523\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">20<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-524\" 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{9 \\pi}{20}\\right)<\/script> = 1<br> Answer:<br> cot<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-39-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-525\" style=\"width: 2.379em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.053em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.447em, 1001.91em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-526\"><span class=\"mrow\" id=\"MathJax-Span-527\"><span class=\"mo\" id=\"MathJax-Span-528\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-529\"><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(3.544em, 1000.42em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-530\" 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, 1000.7em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-531\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">20<\/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 class=\"mo\" id=\"MathJax-Span-532\" 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-39\">\\left(\\frac{\\pi}{20}\\right)<\/script>. cot<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-40-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-533\" 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(1.4em, 1001.96em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-534\"><span class=\"mrow\" id=\"MathJax-Span-535\"><span class=\"mo\" id=\"MathJax-Span-536\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-537\"><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-538\"><span class=\"mn\" id=\"MathJax-Span-539\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-540\" 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, 1000.7em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-541\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">20<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-542\" 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-40\">\\left(\\frac{3 \\pi}{20}\\right)<\/script>. cot<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-41-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-543\" 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(1.4em, 1001.96em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-544\"><span class=\"mrow\" id=\"MathJax-Span-545\"><span class=\"mo\" id=\"MathJax-Span-546\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-547\"><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-548\"><span class=\"mn\" id=\"MathJax-Span-549\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span class=\"mi\" id=\"MathJax-Span-550\" 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, 1000.7em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-551\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">20<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-552\" 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-41\">\\left(\\frac{5 \\pi}{20}\\right)<\/script>. cot<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-42-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-553\" 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(1.4em, 1001.96em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-554\"><span class=\"mrow\" id=\"MathJax-Span-555\"><span class=\"mo\" id=\"MathJax-Span-556\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-557\"><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-558\"><span class=\"mn\" id=\"MathJax-Span-559\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">7<\/span><span class=\"mi\" id=\"MathJax-Span-560\" 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, 1000.7em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-561\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">20<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-562\" 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-42\">\\left(\\frac{7 \\pi}{20}\\right)<\/script>. cot<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-43-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-563\" 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(1.4em, 1001.96em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-564\"><span class=\"mrow\" id=\"MathJax-Span-565\"><span class=\"mo\" id=\"MathJax-Span-566\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-567\"><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-568\"><span class=\"mn\" id=\"MathJax-Span-569\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">9<\/span><span class=\"mi\" id=\"MathJax-Span-570\" 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, 1000.7em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-571\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">20<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-572\" 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-43\">\\left(\\frac{9 \\pi}{20}\\right)<\/script><br> = cot 9\u00b0. cot 27\u00b0. cot 45\u00b0. cot 63\u00b0. cot 81\u00b0<br> = cot 9\u00b0. cot 27\u00b0. 1.cot (90 \u2013 27) . cot (90 -9)<br> = cot 9\u00b0. cot 27\u00b0. 1. tan 27\u00b0. tan 9\u00b0<br> = 1<\/p>\n\n\n\n<h3>Question 2.<br> (i) Simplify <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-44-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-573\" style=\"width: 12.025em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 10.347em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(0.468em, 1010.35em, 3.59em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-574\"><span class=\"mfrac\" id=\"MathJax-Span-575\"><span style=\"display: inline-block; position: relative; width: 10.068em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.078em, 1009.84em, 4.616em, -999.998em); top: -4.844em; left: 50%; margin-left: -4.984em;\"><span class=\"mrow\" id=\"MathJax-Span-576\"><span class=\"mi\" id=\"MathJax-Span-577\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-578\" style=\"font-size: 70.7%;\"><\/span><span class=\"mrow\" id=\"MathJax-Span-579\"><span class=\"mo\" id=\"MathJax-Span-580\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">(<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-581\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-582\"><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(3.544em, 1000.79em, 4.15em, -999.998em); top: -4.331em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-583\"><span class=\"mn\" id=\"MathJax-Span-584\" style=\"font-size: 50%; font-family: MathJax_Main;\">11<\/span><span class=\"mi\" id=\"MathJax-Span-585\" style=\"font-size: 50%; 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(3.544em, 1000.24em, 4.15em, -999.998em); top: -3.679em; left: 50%; margin-left: -0.137em;\"><span class=\"mn\" id=\"MathJax-Span-586\" style=\"font-size: 50%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.209em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-587\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-588\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">tan<\/span><span class=\"mo\" id=\"MathJax-Span-589\" style=\"font-size: 70.7%;\"><\/span><span class=\"mrow\" id=\"MathJax-Span-590\"><span class=\"mo\" id=\"MathJax-Span-591\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">(<\/span><\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-592\"><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(3.544em, 1000.79em, 4.15em, -999.998em); top: -4.331em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-593\"><span class=\"mn\" id=\"MathJax-Span-594\" style=\"font-size: 50%; font-family: MathJax_Main;\">35<\/span><span class=\"mi\" id=\"MathJax-Span-595\" style=\"font-size: 50%; 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(3.544em, 1000.24em, 4.15em, -999.998em); top: -3.679em; left: 50%; margin-left: -0.137em;\"><span class=\"mn\" id=\"MathJax-Span-596\" style=\"font-size: 50%; font-family: MathJax_Main;\">6<\/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: -1.209em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-597\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-598\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">sec<\/span><span class=\"mo\" id=\"MathJax-Span-599\" style=\"font-size: 70.7%;\"><\/span><span class=\"mrow\" id=\"MathJax-Span-600\"><span class=\"mo\" id=\"MathJax-Span-601\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">(<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-602\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mfrac\" id=\"MathJax-Span-603\"><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(3.544em, 1000.56em, 4.15em, -999.998em); top: -4.331em; left: 50%; margin-left: -0.277em;\"><span class=\"mrow\" id=\"MathJax-Span-604\"><span class=\"mn\" id=\"MathJax-Span-605\" style=\"font-size: 50%; font-family: MathJax_Main;\">7<\/span><span class=\"mi\" id=\"MathJax-Span-606\" style=\"font-size: 50%; 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(3.544em, 1000.24em, 4.15em, -999.998em); top: -3.679em; left: 50%; margin-left: -0.137em;\"><span class=\"mn\" id=\"MathJax-Span-607\" style=\"font-size: 50%; 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, 1000.65em, 1.214em, -999.998em); top: -1.209em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.655em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-608\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.078em, 1009em, 4.616em, -999.998em); top: -3.306em; left: 50%; margin-left: -4.564em;\"><span class=\"mrow\" id=\"MathJax-Span-609\"><span class=\"mi\" id=\"MathJax-Span-610\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">cot<\/span><span class=\"mo\" id=\"MathJax-Span-611\" style=\"font-size: 70.7%;\"><\/span><span class=\"mrow\" id=\"MathJax-Span-612\"><span class=\"mo\" id=\"MathJax-Span-613\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">(<\/span><\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-614\"><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(3.544em, 1000.56em, 4.15em, -999.998em); top: -4.331em; left: 50%; margin-left: -0.277em;\"><span class=\"mrow\" id=\"MathJax-Span-615\"><span class=\"mn\" id=\"MathJax-Span-616\" style=\"font-size: 50%; font-family: MathJax_Main;\">5<\/span><span class=\"mi\" id=\"MathJax-Span-617\" style=\"font-size: 50%; 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(3.544em, 1000.24em, 4.15em, -999.998em); top: -3.679em; left: 50%; margin-left: -0.137em;\"><span class=\"mn\" id=\"MathJax-Span-618\" style=\"font-size: 50%; 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, 1000.65em, 1.214em, -999.998em); top: -1.209em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.655em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-619\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><span class=\"texatom\" id=\"MathJax-Span-620\"><span class=\"mrow\" id=\"MathJax-Span-621\"><span class=\"mi\" id=\"MathJax-Span-622\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">c<\/span><span class=\"mi\" id=\"MathJax-Span-623\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">o<\/span><span class=\"mi\" id=\"MathJax-Span-624\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">s<\/span><span class=\"mi\" id=\"MathJax-Span-625\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">e<\/span><span class=\"mi\" id=\"MathJax-Span-626\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-627\"><span class=\"mo\" id=\"MathJax-Span-628\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">(<\/span><\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-629\"><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(3.544em, 1000.56em, 4.15em, -999.998em); top: -4.331em; left: 50%; margin-left: -0.277em;\"><span class=\"mrow\" id=\"MathJax-Span-630\"><span class=\"mn\" id=\"MathJax-Span-631\" style=\"font-size: 50%; font-family: MathJax_Main;\">7<\/span><span class=\"mi\" id=\"MathJax-Span-632\" style=\"font-size: 50%; 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(3.544em, 1000.24em, 4.15em, -999.998em); top: -3.679em; left: 50%; margin-left: -0.137em;\"><span class=\"mn\" id=\"MathJax-Span-633\" style=\"font-size: 50%; 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, 1000.65em, 1.214em, -999.998em); top: -1.209em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.655em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-634\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-635\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-636\" style=\"font-size: 70.7%;\"><\/span><span class=\"mrow\" id=\"MathJax-Span-637\"><span class=\"mo\" id=\"MathJax-Span-638\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">(<\/span><\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-639\"><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(3.544em, 1000.79em, 4.15em, -999.998em); top: -4.331em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-640\"><span class=\"mn\" id=\"MathJax-Span-641\" style=\"font-size: 50%; font-family: MathJax_Main;\">17<\/span><span class=\"mi\" id=\"MathJax-Span-642\" style=\"font-size: 50%; 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(3.544em, 1000.24em, 4.15em, -999.998em); top: -3.679em; left: 50%; margin-left: -0.137em;\"><span class=\"mn\" id=\"MathJax-Span-643\" style=\"font-size: 50%; font-family: MathJax_Main;\">6<\/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: -1.209em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-644\" style=\"vertical-align: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Size2;\">)<\/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, 1010.07em, 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: 10.068em; 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: -1.403em; border-left: 0px solid; width: 0px; height: 3.408em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-44\">\\frac{\\sin \\left(-\\frac{11 \\pi}{3}\\right) \\tan \\left(\\frac{35 \\pi}{6}\\right) \\sec \\left(-\\frac{7 \\pi}{3}\\right)}{\\cot \\left(\\frac{5 \\pi}{4}\\right) {cosec}\\left(\\frac{7 \\pi}{4}\\right) \\cos \\left(\\frac{17 \\pi}{6}\\right)}<\/script><\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7419\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6a-8.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a) 8\" width=\"322\" height=\"798\" sizes=\"auto, (max-width: 322px) 100vw, 322px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a) 8\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a)\"><\/p>\n\n\n\n<p>(ii) If tan 20 \u00b0 = p, prove that<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-45-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-645\" style=\"width: 10.953em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 9.415em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.074em, 1009.41em, 3.031em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-646\"><span class=\"mfrac\" id=\"MathJax-Span-647\"><span style=\"display: inline-block; position: relative; width: 5.827em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1005.69em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -2.84em;\"><span class=\"mrow\" id=\"MathJax-Span-648\"><span class=\"mi\" id=\"MathJax-Span-649\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">tan<\/span><span class=\"mo\" id=\"MathJax-Span-650\" style=\"font-size: 70.7%;\"><\/span><span class=\"msubsup\" id=\"MathJax-Span-651\" 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-652\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">610<\/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-653\"><span class=\"mrow\" id=\"MathJax-Span-654\"><span class=\"mo\" id=\"MathJax-Span-655\" style=\"font-size: 50%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-656\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-657\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">tan<\/span><span class=\"mo\" id=\"MathJax-Span-658\" style=\"font-size: 70.7%;\"><\/span><span class=\"msubsup\" id=\"MathJax-Span-659\" 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-660\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">700<\/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-661\"><span class=\"mrow\" id=\"MathJax-Span-662\"><span class=\"mo\" id=\"MathJax-Span-663\" 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(3.357em, 1005.69em, 4.196em, -999.998em); top: -3.586em; left: 50%; margin-left: -2.84em;\"><span class=\"mrow\" id=\"MathJax-Span-664\"><span class=\"mi\" id=\"MathJax-Span-665\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">tan<\/span><span class=\"mo\" id=\"MathJax-Span-666\" style=\"font-size: 70.7%;\"><\/span><span class=\"msubsup\" id=\"MathJax-Span-667\" 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-668\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">560<\/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-669\"><span class=\"mrow\" id=\"MathJax-Span-670\"><span class=\"mo\" id=\"MathJax-Span-671\" style=\"font-size: 50%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-672\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-673\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">tan<\/span><span class=\"mo\" id=\"MathJax-Span-674\" style=\"font-size: 70.7%;\"><\/span><span class=\"msubsup\" id=\"MathJax-Span-675\" 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-676\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">470<\/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-677\"><span class=\"mrow\" id=\"MathJax-Span-678\"><span class=\"mo\" id=\"MathJax-Span-679\" style=\"font-size: 50%; font-family: MathJax_Main;\">\u2218<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1005.83em, 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: 5.827em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-680\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-681\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.68em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.264em, 1001.59em, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.79em;\"><span class=\"mrow\" id=\"MathJax-Span-682\"><span class=\"mn\" id=\"MathJax-Span-683\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-684\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"msubsup\" id=\"MathJax-Span-685\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.38em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-686\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">p<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.285em; left: 0.375em;\"><span class=\"mn\" id=\"MathJax-Span-687\" 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><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1001.59em, 4.289em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.79em;\"><span class=\"mrow\" id=\"MathJax-Span-688\"><span class=\"mn\" id=\"MathJax-Span-689\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-690\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"msubsup\" id=\"MathJax-Span-691\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.38em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-692\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">p<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.192em; left: 0.375em;\"><span class=\"mn\" id=\"MathJax-Span-693\" 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><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.68em, 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: 1.68em; 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: -0.7em; border-left: 0px solid; width: 0px; height: 2.057em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-45\">\\frac{\\tan 610^{\\circ}+\\tan 700^{\\circ}}{\\tan 560^{\\circ}-\\tan 470^{\\circ}}=\\frac{1-p^2}{1+p^2}<\/script><br> Answer:<br> Given that tan 20\u00b0 = p then<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7418\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6a-9.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a) 9\" width=\"313\" height=\"291\" sizes=\"auto, (max-width: 313px) 100vw, 313px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a) 9\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a)\"><\/p>\n\n\n\n<p>(iii) If \u03b1, \u03b2 are complementary angles such that b sin \u03b1 = a, then find the value of (sin \u03b1 cos \u03b2 \u2013 cos \u03b1 sin \u03b2)<br> Answer:<br> Given \u03b1, \u03b2 are complementary angles \u03b1 + \u03b2 = 90\u00b0<br> \u21d2 \u03b2 = 90\u00b0 \u2013 \u03b1<br> \u2234 sin \u03b1 cos \u03b2 \u2013 cos \u03b1 sin \u03b2<br> = sin (\u03b1 \u2013 \u03b2)<br> = sin[\u03b1 \u2013 (90 \u2013 \u03b1)]<br> = sin [2\u03b1 \u2013 90\u00b0]<br> = -sin[90 \u2013 2\u03b1]<br> = -cos 2\u03b1<br> = -(1 \u2013 2sin<sup>2<\/sup>\u03b1) = -1 + 2sin<sup>2<\/sup>\u03b1<br> = -1 + 2<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-46-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-694\" style=\"width: 2.658em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.286em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.866em, 1002.1em, 3.963em, -999.998em); top: -3.166em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-695\"><span class=\"mrow\" id=\"MathJax-Span-696\"><span class=\"mo\" id=\"MathJax-Span-697\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-698\"><span style=\"display: inline-block; position: relative; width: 0.795em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.264em, 1000.7em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.324em;\"><span class=\"msubsup\" id=\"MathJax-Span-699\"><span style=\"display: inline-block; position: relative; width: 0.701em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.38em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-700\" 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; top: -4.285em; left: 0.375em;\"><span class=\"mn\" id=\"MathJax-Span-701\" 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 style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.61em, 4.15em, -999.998em); top: -3.493em; left: 50%; margin-left: -0.324em;\"><span class=\"msubsup\" id=\"MathJax-Span-702\"><span style=\"display: inline-block; position: relative; width: 0.608em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.28em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-703\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">b<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.285em; left: 0.329em;\"><span class=\"mn\" id=\"MathJax-Span-704\" 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 style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.79em, 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: 0.795em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-705\" 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-46\">\\left(\\frac{a^2}{b^2}\\right)<\/script><br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-47-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-706\" style=\"width: 3.031em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.612em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.074em, 1002.61em, 2.938em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-707\"><span class=\"mfrac\" id=\"MathJax-Span-708\"><span style=\"display: inline-block; position: relative; width: 2.332em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.218em, 1002.19em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -1.069em;\"><span class=\"mrow\" id=\"MathJax-Span-709\"><span class=\"mn\" id=\"MathJax-Span-710\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"msubsup\" id=\"MathJax-Span-711\"><span style=\"display: inline-block; position: relative; width: 0.701em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.38em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-712\" 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; top: -4.285em; left: 0.375em;\"><span class=\"mn\" id=\"MathJax-Span-713\" 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-714\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"msubsup\" id=\"MathJax-Span-715\"><span style=\"display: inline-block; position: relative; width: 0.608em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.28em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-716\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">b<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.285em; left: 0.329em;\"><span class=\"mn\" id=\"MathJax-Span-717\" 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><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.61em, 4.15em, -999.998em); top: -3.493em; left: 50%; margin-left: -0.324em;\"><span class=\"msubsup\" id=\"MathJax-Span-718\"><span style=\"display: inline-block; position: relative; width: 0.608em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.28em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-719\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">b<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.285em; left: 0.329em;\"><span class=\"mn\" id=\"MathJax-Span-720\" 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 style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1002.33em, 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: 2.332em; 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: 2.003em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-47\">\\frac{2 a^2-b^2}{b^2}<\/script><\/p>\n\n\n\n<h3>Question 3.<br> (i) If cos A = cos B = \u2013 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-48-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-721\" 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-722\"><span class=\"mfrac\" id=\"MathJax-Span-723\"><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-724\" 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-725\" 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-48\">\\frac{1}{2}<\/script>, A does not lie in the second quadrant and B does not lie in third quadrant, then find the value of <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-49-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-726\" style=\"width: 6.107em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.268em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.167em, 1005.27em, 2.891em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-727\"><span class=\"mfrac\" id=\"MathJax-Span-728\"><span style=\"display: inline-block; position: relative; width: 4.988em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1004.85em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -2.421em;\"><span class=\"mrow\" id=\"MathJax-Span-729\"><span class=\"mn\" id=\"MathJax-Span-730\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span class=\"mi\" id=\"MathJax-Span-731\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-732\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-733\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">B<\/span><span class=\"mo\" id=\"MathJax-Span-734\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-735\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"mi\" id=\"MathJax-Span-736\" style=\"font-size: 70.7%; font-family: MathJax_Main; padding-left: 0.235em;\">tan<\/span><span class=\"mo\" id=\"MathJax-Span-737\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-738\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">A<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1003.82em, 4.196em, -999.998em); top: -3.586em; left: 50%; margin-left: -1.908em;\"><span class=\"mrow\" id=\"MathJax-Span-739\"><span class=\"mi\" id=\"MathJax-Span-740\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">tan<\/span><span class=\"mo\" id=\"MathJax-Span-741\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-742\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic; padding-left: 0.235em;\">B<\/span><span class=\"mo\" id=\"MathJax-Span-743\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-744\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-745\" style=\"font-size: 70.7%;\"><\/span><span class=\"mi\" id=\"MathJax-Span-746\" 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, 1004.99em, 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: 4.988em; 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-49\">\\frac{4 \\sin B-3 \\tan A}{\\tan B+\\sin A}<\/script><\/h3>\n\n<p>Answer:<br> cos A = \u2013<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-50-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-747\" 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-748\"><span class=\"mfrac\" id=\"MathJax-Span-749\"><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-750\" 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-751\" 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-50\">\\frac{1}{2}<\/script> and A does not lie in second quadrant<br> \u21d2 A lies in third quadrant<br> cos B = \u2013<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-51-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-752\" 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-753\"><span class=\"mfrac\" id=\"MathJax-Span-754\"><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-755\" 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-756\" 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-51\">\\frac{1}{2}<\/script> and B does not lie in third quadrant<br> \u21d2 B lies in second quadrant<br> cos A = \u2013<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-52-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-757\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-758\"><span class=\"mfrac\" id=\"MathJax-Span-759\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-760\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">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-761\" 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-52\">\\frac{1}{2}<\/script> and A lie in third quadrant<br> \u21d2 A = 240\u00b0<br> cos B = \u2013<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-53-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-762\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-763\"><span class=\"mfrac\" id=\"MathJax-Span-764\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-765\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">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-766\" 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-53\">\\frac{1}{2}<\/script> and B lies in second quadrant.<br> \u21d2 B = 120\u00b0<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7417\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6a-10.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a) 10\" width=\"311\" height=\"294\" sizes=\"auto, (max-width: 311px) 100vw, 311px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a) 10\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a)\"><\/p>\n\n\n\n<p>(ii) If 8 tan A = -15 and 25 sin B = -7 and neither A nor B is in the fourth quadrant, then show that sin A cos B + cos A sin B = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-54-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-767\" style=\"width: 2.332em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.006em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1002.01em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-768\"><span class=\"mfrac\" id=\"MathJax-Span-769\"><span style=\"display: inline-block; position: relative; width: 1.726em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.59em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -0.79em;\"><span class=\"mrow\" id=\"MathJax-Span-770\"><span class=\"mo\" id=\"MathJax-Span-771\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-772\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">304<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1001.03em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.51em;\"><span class=\"mn\" id=\"MathJax-Span-773\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">425<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.73em, 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: 1.726em; 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-54\">\\frac{-304}{425}<\/script><br> Answer:<br> 8 tan A = -15 25 sin B = -7<br> \u21d2 tan A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-55-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-774\" 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(1.214em, 1001.68em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-775\"><span class=\"mfrac\" id=\"MathJax-Span-776\"><span style=\"display: inline-block; position: relative; width: 1.4em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.21em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -0.603em;\"><span class=\"mrow\" id=\"MathJax-Span-777\"><span class=\"mo\" id=\"MathJax-Span-778\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-779\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">15<\/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-780\" 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, 1001.4em, 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: -0.484em; border-left: 0px solid; width: 0px; height: 1.624em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-55\">\\frac{-15}{8}<\/script> \u21d2 sin B = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-56-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-781\" style=\"width: 1.54em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.307em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.214em, 1001.31em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-782\"><span class=\"mfrac\" id=\"MathJax-Span-783\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.89em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -0.464em;\"><span class=\"mrow\" id=\"MathJax-Span-784\"><span class=\"mo\" id=\"MathJax-Span-785\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-786\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">7<\/span><\/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-787\" 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, 1001.03em, 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: 1.027em; 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-56\">\\frac{-7}{25}<\/script><br> Given neither A nor B is in the fourth quadrant, clearly A is in second quadrant and B is in third quadrant,<br> sin A cos B + cos A sin B<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7416\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6a-11.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(a) 11\" width=\"269\" height=\"104\" data-pin-nopin=\"true\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>I. Question 1. Convert the following into simplest form (i) tan (\u03b8 \u2013 14\u03c0) Answer: tan (\u03b8 \u2013 14\u03c0) = tan [- (14\u03c0 \u2013 \u03b8)] = \u2013 tan (14\u03c0 \u2013 \u03b8) = \u2013 tan [ 2(7\u03c0) \u2013 \u03b8) = \u2013 tan (-\u03b8) = tan \u03b8 (ii) cot (21\u03c02 \u2013 \u03b8) Answer: cot (21\u03c02 \u2013 \u03b8) [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":3353,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[312,15],"tags":[],"class_list":{"0":"post-3066","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-inter-telangana","8":"category-telangana"},"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(a)<\/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-6a\/\" \/>\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(a)\" \/>\n<meta property=\"og:description\" content=\"I. Question 1. Convert the following into simplest form (i) tan (\u03b8 \u2013 14\u03c0) Answer: tan (\u03b8 \u2013 14\u03c0) = tan [- (14\u03c0 \u2013 \u03b8)] = \u2013 tan (14\u03c0 \u2013 \u03b8) = \u2013 tan [ 2(7\u03c0) \u2013 \u03b8) = \u2013 tan (-\u03b8) = tan \u03b8 (ii) cot (21\u03c02 \u2013 \u03b8) Answer: cot (21\u03c02 \u2013 \u03b8) [&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-6a\/\" \/>\n<meta property=\"og:site_name\" content=\"Manabadi Boards\" \/>\n<meta property=\"article:published_time\" content=\"2026-01-30T09:03:10+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2026-01-30T09:03:14+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/boardscdn.sgp1.digitaloceanspaces.com\/boards\/wp-content\/uploads\/2026\/01\/07105949\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6a.jpg\" 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Question 1. Convert the following into simplest form (i) tan (\u03b8 \u2013 14\u03c0) Answer: tan (\u03b8 \u2013 14\u03c0) = tan [- (14\u03c0 \u2013 \u03b8)] = \u2013 tan (14\u03c0 \u2013 \u03b8) = \u2013 tan [ 2(7\u03c0) \u2013 \u03b8) = \u2013 tan (-\u03b8) = tan \u03b8 (ii) cot (21\u03c02 \u2013 \u03b8) Answer: cot (21\u03c02 \u2013 \u03b8) [&hellip;]","og_url":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6a\/","og_site_name":"Manabadi Boards","article_published_time":"2026-01-30T09:03:10+00:00","article_modified_time":"2026-01-30T09:03:14+00:00","og_image":[{"width":900,"height":600,"url":"https:\/\/boardscdn.sgp1.digitaloceanspaces.com\/boards\/wp-content\/uploads\/2026\/01\/07105949\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6a.jpg","type":"image\/jpeg"}],"author":"Junaid","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Junaid","Est. reading 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