Example 5 - Express tan-1 cos⁡x/(1 - sin⁡x) - Chapter 2 Inverse - Not clear how to approach

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Example 5 Express tan-1 cos⁡x/(1 − sin⁡x ) , - π/2 < x < 3π/2 in the simplest form Lets first calculate cos x & 1 – sin x Solving tan-1 (cos⁡x/(1 〖− sin〗⁡x )) = tan-1 [(cos2 x/2 − sin2 x/2)/(1 − (2 〖sin 〗⁡〖x/2 cos⁡〖 x/2〗 〗 ) )] = tan-1 [(cos2 x/2 − sin2 x/2)/(1 − 2 〖sin 〗⁡〖x/2 〖 cos 〗⁡〖x/2〗 〗 )] = tan-1 [(cos2 x/2 − sin2 x/2)/(cos2 x/2 + sin2 x/2 − 2 〖sin 〗⁡〖x/2 cos⁡〖 x/2〗 〗 )] = tan-1 [(cos x/2 + sin x/2)(cos x/2 − sin x/2)/(cos x/2 − sin x/2)^2 ] = tan-1 [((cos x/2 + sin x/2))/((cos x/2 − sin x/2) )] Dividing by cos 𝑥/2 = tan-1 ((cos⁡〖 𝑥/( 2 )〗/〖𝑐𝑜𝑠 〗⁡〖 𝑥/2〗 + sin⁡〖 𝑥/( 2 )〗/〖𝑐𝑜𝑠 〗⁡〖 𝑥/2〗 )/(𝑐𝑜𝑠⁡〖 𝑥/( 2 )〗/〖𝑐𝑜𝑠 〗⁡〖 𝑥/2〗 − 𝑠𝑖𝑛⁡〖 𝑥/( 2 )〗/〖𝑐𝑜𝑠 〗⁡〖 𝑥/2〗 )) = tan-1 [(1 + 〖tan 〗⁡〖𝑥/2〗)/(1 − tan⁡〖 𝑥/2〗 )] = tan-1 [(1 + 〖tan 〗⁡〖𝑥/2〗)/(1 − 〖1. tan〗⁡〖 𝑥/2〗 )] = tan-1 ((𝑡𝑎𝑛⁡〖 𝜋/4〗 + 〖𝑡𝑎𝑛 〗⁡〖𝑥/2〗)/( 1− 〖𝑡𝑎𝑛 〗⁡〖𝜋/4 〗.〖 𝑡𝑎𝑛 〗⁡〖𝑥/2〗 )) = tan -1 [tan⁡(π/4+x/2 ) ] = π/4+x/2

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