Ex 8.3, 4 (viii) - Chapter 8 Class 10 Introduction to Trignometry

Transcript

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