Explanation

(d)
Given, $f(x) = \cos{x - \sin x}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \sqrt{2}\left( \frac{1}{\sqrt{2}}\cos{x - \frac{1}{\sqrt{2}}\sin x} \right)$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \sqrt{2}\cos\left( \frac{\pi}{2} + x \right)$
Since, $- 1 \leq \cos{x \leq 1\ \ \Rightarrow \ \ - 1 \leq \cos{\left( \frac{\pi}{4} + x \right) \leq 1}}$
$\Rightarrow \ \ \ \ \ - \sqrt{2} \leq \sqrt{2}\cos{\left( \frac{\pi}{4} + x \right) \leq \sqrt{2}}$
$\therefore$ Range is $\lbrack - \sqrt{2},\ \sqrt{2}\rbrack$