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Ex 8.3, 4 Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A Solving L.H.S (sin A + cosec A)2 + (cos A + sec A)2 = (sin2 A + cosec2 A + 2sin A cosec A) + (cos2 A + sec2 A + 2 cos A . sec A) = (sin2 A + cosec2 A + 2sin A . 1/sin⁡〖 𝐴〗 ) + (cos2 A + sec2 A + 2 cos A.1/cos⁡〖 𝐴〗 ) = (sin2 A + cosec2 A + 2) + (cos2 A + sec2 A + 2) Using cosec2 A = 1 + cot2 A sec2 A = 1 + tan2 A = (sin2 A + (1 + cot2 A) +2) + (cos2 A + (1 + tan2 A) + 2) = sin2 A + cos2 A + 1 + cot2 A + 2 + 1 + tan2 A + 2 = (sin2 A + cos2 A) + cot2 A + tan2 A + (1 + 2 + 1 + 2) = sin2 A + cot2 A + 1 + 2 + cos2 A + 1 + tan2 A + 2 = (sin2 A + cos2 A) + cot2 A + tan2 A + (1 + 2 + 1 + 2) Using sin2 A + cos2 A = 1 = 1 + cot2 A + tan2 A + 6 = 7 + cot2 A + tan2 A = R.H.S Hence proved = sin2 A + cot2 A + 1 + 2 + cos2 A + 1 + tan2 A + 2 = (sin2 A + cos2 A) + cot2 A + tan2 A + (1 + 2 + 1 + 2) Using sin2 A + cos2 A = 1 = 1 + cot2 A + tan2 A + 6 = 7 + cot2 A + tan2 A = R.H.S Hence proved

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.