{"id":9861,"date":"2024-02-23T01:52:50","date_gmt":"2024-02-22T20:22:50","guid":{"rendered":"https:\/\/tsboardsolutions.in\/?p=9861"},"modified":"2024-02-24T17:57:45","modified_gmt":"2024-02-24T12:27:45","slug":"ts-10th-class-maths-solutions-chapter-11-trigonometry-ex-11-4","status":"publish","type":"post","link":"https:\/\/tsboardsolutions.in\/ts-10th-class-maths-solutions-chapter-11-trigonometry-ex-11-4\/","title":{"rendered":"TS 10th Class Maths Solutions Chapter 11 Trigonometry Ex 11.4"},"content":{"rendered":"

Students can practice\u00a010th Class Maths Study Material Telangana<\/a> Chapter 11 Trigonometry Ex 11.4 to get the best methods of solving problems.<\/p>\n

TS 10th Class Maths Solutions Chapter 11 Trigonometry Exercise 11.4<\/h2>\n

Question 1.
\nEvaluate the following :
\ni) (1 + tan \u03b8 + sec \u03b8) (1 + cot \u03b8 – cosec \u03b8)
\nii) (sin \u03b8 + cos \u03b8)2<\/sup> + (sin \u03b8 – cos \u03b8)2<\/sup>
\niii) (sec2<\/sup> \u03b8 – 1) (cosec2<\/sup> \u03b8 – 1) (AS1<\/sub>)
\nSolution:
\ni) (1 + tan \u03b8 + sec \u03b8) (1 + cot \u03b8 – cosec \u03b8)
\n= (1 + tan \u03b8 + sec \u03b8)
\n\"TS<\/p>\n

ii) (sin \u03b8 + cos \u03b8)2<\/sup> + (sin \u03b8 – cos \u03b8)2<\/sup>
\n(sin \u03b8 + cos \u03b8)2<\/sup> = sin2<\/sup> \u03b8 + cos2<\/sup> \u03b8 + 2 sin \u03b8 cos \u03b8 …………. (1)
\n(sin \u03b8 – cos \u03b8)2<\/sup> = sin2<\/sup> \u03b8 + cos2<\/sup> \u03b8 – 2 sin \u03b8 cos \u03b8 ……………. (2)
\nAdding (1) & (2)
\nsin2<\/sup> \u03b8 + cos2<\/sup> \u03b8 + 2 sin \u03b8 (cos \u03b8) + sin2<\/sup> \u03b8 + cos2<\/sup> \u03b8 – 2 sin \u03b8 (cos \u03b8)
\n= 2 (sin2<\/sup> \u03b8 + cos2<\/sup> \u03b8)
\n= 2(1) = 2<\/p>\n

\"TS<\/p>\n

iii) (sec2<\/sup> \u03b8 – 1) (cosec2<\/sup> \u03b8 – 1)
\nsec2<\/sup> \u03b8 – 1 = tan2<\/sup> \u03b8 …………… (1)
\n[\u2235 1 + tan2<\/sup> \u03b8 = sec2<\/sup> \u03b8 \u21d2 tan2<\/sup> \u03b8 = sec2<\/sup> \u03b8 – 1]
\ncosec2<\/sup> \u03b8 – 1 = cot2<\/sup> \u03b8 …………… (2)
\n[\u2235 1 + cot2<\/sup> \u03b8 = cosec2<\/sup> \u03b8 \u21d2 cot2<\/sup> \u03b8 = cosec2<\/sup> \u03b8 – 1]
\n\u21d2 tan2<\/sup> \u03b8 cot2<\/sup> \u03b8
\n\u21d2 (tan \u03b8 cot \u03b8)2<\/sup> [\u2235 (tan \u03b8) (cot \u03b8) = 1]
\n\u21d2 (1)2<\/sup> = 1<\/p>\n

Question 2.
\nShow that\u00a0 \u00a0(A.P. June ’15)
\n(cosec \u03b8 – cot \u03b8)2<\/sup> = \\(\\frac{1-\\cos \\theta}{1+\\cos \\theta}\\) (AS2<\/sub>)
\nSolution:
\n(cosec \u03b8 – cot \u03b8)2<\/sup> = \\(\\left(\\frac{1}{\\sin \\theta}-\\frac{\\cos \\theta}{\\sin \\theta}\\right)^2\\)
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 3.
\nShow that \\(\\sqrt{\\frac{1+\\sin A}{1-\\sin A}}\\) = sec A + tan A. (AS2<\/sub>)
\nSolution:
\n\"TS<\/p>\n

Question 4.
\nShow that \\(\\frac{1-\\tan ^2 A}{\\cot ^2 A-1}\\) = tan2<\/sup> A. (AS2<\/sub>)
\nSolution:
\n\"TS<\/p>\n

Question 5.
\nShow that \\(\\frac{1}{\\cos \\theta}\\) – cos \u03b8 = tan \u03b8 . sin \u03b8 (AS2<\/sub>)
\nSolution:
\nL.H.S. = \\(\\frac{1}{\\cos \\theta}\\) = cos \u03b8 = \\(\\frac{1-\\cos ^2 \\theta}{\\cos \\theta}\\)
\n= \\(\\frac{\\sin ^2 \\theta}{\\cos \\theta}\\)
\n[\u2235 sin2<\/sup>\u03b8 + cos2<\/sup>\u03b8 = 1 \u21d2 sin2<\/sup>\u03b8 = 1 – cos2<\/sup>\u03b8]
\n= \\(\\frac{\\sin \\theta}{\\cos \\theta}\\) . sin \u03b8
\n= tan \u03b8 . sin \u03b8 (\u2235 tan \u03b8 = \\(\\frac{\\sin \\theta}{\\cos \\theta}\\))<\/p>\n

\"TS<\/p>\n

Question 6.
\nSimplify: sec A (1 – sin A). (sec A + tan A). (AS3<\/sub>)
\nSolution:
\nsec A (1 – sin A) = (sec A – sec A . sin A)
\n= (sec A – \\(\\frac{\\sin A}{\\cos A}\\))
\n= (sec A – tan A)
\n\u2234 sec A (1 – sin A) = (sec A + tan A)
\n= (sec A + tan A) (sec A – tan A)
\n= sec2<\/sup> A – tan2<\/sup> A
\n= 1
\n[\u2235 1 + tan2<\/sup> A = sec2<\/sup> A \u21d2 sec2<\/sup> A – tan2<\/sup> A = 1]<\/p>\n

Question 7.
\nProve that (sin A + cosec A)2<\/sup> + (cos A + sec A)2<\/sup> = 7 + tan2<\/sup> A + cot2<\/sup> A) (AS1<\/sub>, AS2<\/sub>)
\nSolution:
\nL.H.S. = (sin A + cosec A)2<\/sup> + (cos A + sec A)2<\/sup>
\n= sin2<\/sup> A + cosec2<\/sup> A + 2 sin A. cosec A + cos2<\/sup> A + sec2<\/sup> A + 2 cos A . sec A
\n= (sin2<\/sup> A + cos2<\/sup> A) + cosec2<\/sup> A+ 2 + sec2<\/sup> A + 2
\n= 1 + cosec2<\/sup> A + 2 + sec2<\/sup> A + 2
\n[\u2235 sin2<\/sup> A + cos2<\/sup> A = 1]
\n= 5 + cosec2<\/sup> A + sec2<\/sup> A
\n= 5 + 1 + cot2<\/sup> A + 1 + tan2<\/sup> A
\n(\u2235 1 + cot2<\/sup> A = cosec2<\/sup> A and 1 + tan2<\/sup> A = sec2<\/sup> A)
\n= 7 + tan2<\/sup> A + cot2<\/sup> A<\/p>\n

\"TS<\/p>\n

Question 8.
\nSimplify : (T.S. Mar.’15) (AS1<\/sub>, AS3<\/sub>)
\n(1 – cos \u03b8) (1 + cos \u03b8) (1 + cot2<\/sup> \u03b8)
\nSolution:
\n(1 – cos \u03b8) (1 + cos \u03b8) (1 + cot2<\/sup> \u03b8)
\n= (1 – cos2<\/sup> \u03b8) (1 + cot2<\/sup> \u03b8)
\n[\u2235 (a – b) (a + b) = a2<\/sup> – b2<\/sup>]
\n= sin2<\/sup> \u03b8 (1 + cot2<\/sup> \u03b8)
\n(\u2235 sin2<\/sup> \u03b8 + cos2<\/sup> \u03b8 = 1 \u21d2 sin2<\/sup> \u03b8 = 1- cos2<\/sup> \u03b8)
\n= sin2<\/sup> \u03b8 + sin2<\/sup> \u03b8 . cot2<\/sup> \u03b8
\n= sin2<\/sup> \u03b8 + sin2<\/sup> \u03b8 . \\(\\frac{\\cos ^2 \\theta}{\\sin ^2 \\theta}\\)
\n(\u2235 cot \u03b8 = \\(\\frac{\\cos \\theta}{\\sin \\theta}\\) cot2<\/sup>\u03b8 = \\(\\frac{\\cos ^2 \\theta}{\\sin ^2 \\theta}\\))
\n= sin2<\/sup> \u03b8 + cos2<\/sup> \u03b8
\n= 1<\/p>\n

Question 9.
\nIf sec \u03b8 + tan \u03b8 = p, then what is the value of sec \u03b8 – tan \u03b8 ? (AS1<\/sub>)
\nSolution:
\nWe know that 1 + tan2<\/sup> \u03b8 = sec2 \u03b8
\n\u21d2 sec2<\/sup> \u03b8 – tan2<\/sup> \u03b8 = 1
\n\u21d2 (sec \u03b8 + tan \u03b8) (sec \u03b8 – tan \u03b8) = 1
\n[\u2235 a2<\/sup> – b2<\/sup> = (a + b) (a – b)]
\n\u21d2 p(sec \u03b8 – tan \u03b8) = 1
\n\u21d2 sec \u03b8 – tan \u03b8 = \\(\\frac{1}{\\mathrm{p}}\\).
\n\u2234 The value of sec \u03b8 – tan \u03b8 = \\(\\frac{1}{\\mathrm{p}}\\)<\/p>\n

\"TS<\/p>\n

Question 10.
\nIf cosec \u03b8 + cot \u03b8 = k, then prove that cos \u03b8 = \\(\\frac{\\mathrm{k}^2-1}{\\mathrm{k}^2+1}\\). (A.P. Mar.’16) (AS1<\/sub>)
\nSolution:
\nGiven that cosec \u03b8 – cot \u03b8 = k
\n\"TS<\/p>\n","protected":false},"excerpt":{"rendered":"

Students can practice\u00a010th Class Maths Study Material Telangana Chapter 11 Trigonometry Ex 11.4 to get the best methods of solving problems. TS 10th Class Maths Solutions Chapter 11 Trigonometry Exercise 11.4 Question 1. Evaluate the following : i) (1 + tan \u03b8 + sec \u03b8) (1 + cot \u03b8 – cosec \u03b8) ii) (sin \u03b8 … Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[16],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/9861"}],"collection":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/comments?post=9861"}],"version-history":[{"count":1,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/9861\/revisions"}],"predecessor-version":[{"id":9893,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/9861\/revisions\/9893"}],"wp:attachment":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/media?parent=9861"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/categories?post=9861"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/tags?post=9861"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}