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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. (ii) "cos A" /"1 + sin A" +"1 + sin A" /"cos A" =2 sec A Taking L.H.S (cos⁑ 𝐴)/(1 + sin⁑〖 𝐴〗 )+(1 + sin⁑ 𝐴)/(cos⁑ 𝐴) = (cos⁑ 𝐴 (cos⁑ 𝐴) + (1 + sin⁑ 𝐴)(1 + sin⁑〖 𝐴)γ€—)/((1 + sin⁑ 𝐴)(cos⁑ 𝐴)) = (π’„π’π’”πŸπ‘¨ + (𝟏 + 𝐬𝐒𝐧⁑𝑨 )𝟐)/((𝟏 + π’”π’Šπ’β‘ 𝑨)(𝒄𝒐𝒔⁑𝑨)) = (π‘π‘œπ‘ 2 𝐴 + 1^2 + 𝑠𝑖𝑛2 𝐴 + 2 sin⁑𝐴)/((1 + sin⁑ 𝐴)(cos⁑𝐴)) = ((π’„π’π’”πŸ 𝑨 + π’”π’Šπ’πŸ 𝑨) + 1 + 2 sin⁑ 𝐴)/((1 + sin⁑〖 𝐴)(cos⁑ 𝐴)γ€— ) As cos2 A + sin2 A = 1 = (𝟏 + 1 + 2 sin⁑𝐴)/((1 + sin⁑〖 𝐴)(cos⁑ 𝐴)γ€— ) = (2 + 2 sin⁑ 𝐴)/((1 + sin⁑〖 𝐴)(cos⁑ 𝐴)γ€— ) = (2(1+ sin⁑ 𝐴))/((1 + sin⁑〖 𝐴)(cos⁑ 𝐴)γ€— ) = 𝟐/(𝒄𝒐𝒔⁑ 𝑨) = 2 Γ—1/cos⁑〖 𝐴〗 = 2 sec A = R.H.S ∴ L.H.S = 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 and Computer Science at Teachoo