Differentiate w.r.t. x:
`(i)tan^(-1){sqrt((1+cosx)/(1-cosx))}" "(ii)tan^(-1){sqrt((1+sinx)/(1-sinx))}`

Explanation

To differentiate the given functions with respect to \( x \), we will tackle each part separately.

### Part (i): Differentiate \( y = \tan^{-1}\left(\sqrt{\frac{1 + \cos x}{1 - \cos x}}\right) \)

1. **Simplify the expression inside the inverse tangent**:
\[ \sqrt{\frac{1 + \cos x}{1 - \cos x}} = \sqrt{\frac{1 + \cos x}{\sin^2\left(\frac{x}{2}\right)}} = \frac{\sqrt{1 + \cos x}}{\sin\left(\frac{x}{2}\right)} \]
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