Express each of the following in the simplest form: `tan^(-1){sqrt((1-cosx)/(1+cosx))},\ -pi

Explanation

To express the given functions in their simplest form, we will solve each part step by step.

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

**Step 1:** Use the identities for \(1 - \cos x\) and \(1 + \cos x\).

We know:
\[
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