Prove that `tan^-1((cosx)/(1+sinx))=pi/4-x/2,\ x in (-pi/2,pi/2)`

Explanation

To prove that

\[ \tan^{-1}\left(\frac{\cos x}{1 + \sin x}\right) = \frac{\pi}{4} - \frac{x}{2}, \quad x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right), \]

we will follow these steps:

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