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

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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. "cos A – sin A + 1" /"cos A + sin A – 1" = cosec A + cot A, using the identity cosec2 A = 1 + cot2 A. Solving L.H.S (cos⁡𝐴 − sin⁡𝐴 + 1)/(cos⁡𝐴 + sin⁡𝐴 − 1) Since we need to use cosec and cot identity Dividing both numerator and denominator by sin A = (𝟏/𝒔𝒊𝒏⁡〖 𝑨〗 (cos⁡〖 𝐴 − sin⁡〖𝐴 + 1〗 〗 ))/(𝟏/𝒔𝒊𝒏⁡〖 𝑨〗 (cos⁡〖 𝐴 + sin⁡〖 𝐴 − 1〗 〗 ) ) = (cos⁡〖 𝐴〗/sin⁡〖 𝐴〗 − sin⁡〖 𝐴〗/sin⁡〖 𝐴〗 + 1/sin⁡〖 𝐴〗 )/(cos⁡〖 𝐴〗/sin⁡〖 𝐴〗 + sin⁡〖 𝐴〗/sin⁡〖 𝐴〗 − 1/sin⁡〖 𝐴〗 ) = cot⁡〖 𝐴 − 1 + 𝑐𝑜𝑠𝑒𝑐 𝐴〗/cot⁡〖 𝐴 + 1 − 𝑐𝑜𝑠𝑒𝑐 𝐴〗 = ((cot⁡〖 𝐴 + 𝑐𝑜𝑠𝑒𝑐 𝐴) − 𝟏〗)/((cot⁡〖 𝐴 + 1 − 𝑐𝑜𝑠𝑒𝑐 𝐴) 〗 ) = ((co𝑡⁡〖 𝐴 + 𝑐𝑜𝑠𝑒𝑐 𝐴) − (𝒄𝒐𝒔𝒆𝒄𝟐 𝑨 − 𝒄𝒐𝒕𝟐 𝑨)〗)/((cot⁡ 𝐴 + 1 − 𝑐𝑜𝑠𝑒𝑐 𝐴)) = ((co𝑡⁡〖 𝐴 + 𝑐𝑜𝑠𝑒𝑐 𝐴) − (𝐜𝐨𝒔𝒆𝒄⁡𝑨 − 𝒄𝒐𝒕 𝑨)(𝐜𝐨𝒔𝒆𝒄⁡𝑨 + 𝒄𝒐𝒕 𝑨)〗)/((cot⁡ 𝐴 + 1 − 𝑐𝑜𝑠𝑒𝑐 𝐴)) = ((co𝑡⁡〖 𝐴 + 𝑐𝑜𝑠𝑒𝑐 𝐴) [𝟏 − (𝒄𝒐𝒔𝒆𝒄 𝑨 − 𝒄𝒐𝒕 𝑨 )]〗)/([cot⁡ 𝐴 + 1 − 𝑐𝑜𝑠𝑒𝑐 𝐴]) = ((co𝑡⁡〖 𝐴 + 𝑐𝑜𝑠𝑒𝑐 𝐴) [1 − 𝑐𝑜𝑠𝑒𝑐 𝐴 + 𝑐𝑜𝑡 𝐴]〗)/([cot⁡ 𝐴 + 1 − 𝑐𝑜𝑠𝑒𝑐 𝐴]) = ((co𝑡⁡〖 𝐴 + 𝑐𝑜𝑠𝑒𝑐 𝐴)[cot⁡ 𝐴 + 1 − 𝑐𝑜𝑠𝑒𝑐 𝐴]〗)/([cot⁡ 𝐴 + 1 − 𝑐𝑜𝑠𝑒𝑐 𝐴]) = cot A + cosec A = R.H.S Hence proved