Explanation

(a)
Given,$= \tan^{- 1}{\ \sqrt{\frac{1 - \sin x}{1 + \sin x}}}$
$= \tan^{- 1}\sqrt{\frac{1 - \cos\left( \frac{\pi}{2} - x \right)}{1 + \cos\left( \frac{\pi}{2} - x \right)}}$
$= \tan^{- 1}\left| \tan\left( \frac{\pi}{4} - \frac{x}{2} \right) \right|\$
$= \frac{\pi}{4} - \frac{x}{2}$ $\left\lbrack \because x = \frac{\pi}{6} \right\rbrack$
$\Rightarrow \ \frac{dy}{dx} = \ - \frac{1}{2}$