`tan^(-1)[(cosx)/(1+sinx)]` is equal to `pi/4-x/2,forx in (-pi/2,(3pi)/2)` `pi/4-x/2,forx in (-pi/2,pi/2)` `pi/4-x/2,forx in (-pi/2,pi/2)` `pi/4-x/2,forx in (-(3pi)/2,pi/2)`

Explanation

To solve the problem \( \tan^{-1}\left(\frac{\cos x}{1 + \sin x}\right) \) and show that it is equal to \( \frac{\pi}{4} - \frac{x}{2} \) for the specified ranges, we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ y = \tan^{-1}\left(\frac{\cos x}{1 + \sin x}\right) \]

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