If `y=cot^(-1)(sqrt(cos"x"))-tan^(-1)(sqrt(cos"x")),` prove that `siny=tan^2x/2`

Explanation

To prove that \( \sin y = \tan^2 \frac{x}{2} \) given that \( y = \cot^{-1}(\sqrt{\cos x}) - \tan^{-1}(\sqrt{\cos x}) \), we will follow these steps:

### Step 1: Rewrite the expression for \( y \)
We start with the expression:
\[ y = \cot^{-1}(\sqrt{\cos x}) - \tan^{-1}(\sqrt{\cos x}) \]
Using the identity \( \cot^{-1}(a) = \tan^{-1}\left(\frac{1}{a}\right) \), we can rewrite \( \cot^{-1}(\sqrt{\cos x}) \) as:
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