Explanation

(d)
We have, $y = \tan^{- 1}{(\sec{x - \tan{x)}}}$
$\Rightarrow \frac{dy}{dx} = \frac{d}{dx}\tan^{- 1}\ \left( \frac{1 - \sin x}{\cos x} \right)$
$\Rightarrow \frac{dy}{dx} = \frac{d}{dx}\tan^{- 1}\ \left( \frac{\cos{\frac{x}{2} - \sin\frac{x}{2}}}{\cos{\frac{x}{2} + \sin\frac{x}{2}}} \right)$
$= \frac{d}{dx}\tan^{- 1}\left\lbrack \tan\left( \frac{\pi}{4} - \frac{x}{2} \right) \right\rbrack$
$= \frac{d}{dx}\left( \frac{\pi}{4} - \frac{x}{2} \right) = - \frac{1}{2}$