Explanation

(a)
Let $f(x) = \sqrt{3}\cos{x + \sin x}$
$\Rightarrow f(x) = 2\left( \frac{\sqrt{3}}{2}\cos{x + \frac{1}{2}\sin x} \right) = 2\sin\left( x + \frac{\pi}{3} \right)$
Since, $- 1 \leq \sin{\left( x + \frac{\pi}{3} \right) \leq 1}$
Hence, $f(x)$is maximum$,\$if$\ x + \frac{\pi}{3} = \frac{\pi}{2}$
$\Rightarrow x = \frac{\pi}{6} = 30{^\circ}$