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

Explanation

To prove that \( \tan^{-1}\left(\frac{\cos x}{1+\sin x}\right) = \frac{\pi}{4} - \frac{x}{2} \) for \( x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), we will follow these steps:

### Step 1: Rewrite the Left-Hand Side
Let \( P = \frac{\cos x}{1 + \sin x} \).

### Step 2: Use Trigonometric Identities
We can express \( \cos x \) and \( \sin x \) in terms of half-angle formulas:
- \( \cos x = \cos^2\left(\frac{x}{2}\right) - \sin^2\left(\frac{x}{2}\right) \)
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