Explanation

(d)
Given, $\cos{\theta + \sin{2\theta = 0}}$
$\Rightarrow \cos{\theta + 2\sin{\theta\cos{\theta = 0}}}$
$\Rightarrow \ \cos{\theta\left( 1 + 2\sin\theta \right) = 0}$
$\Rightarrow \ \cos{\theta = 0}$ or $\sin{\theta = - \frac{1}{2}}$
For $\theta \in \lbrack - \pi,\ \pi\rbrack$
$\theta = \frac{\pi}{2},\ - \frac{\pi}{2}$
Or $\theta = - \frac{\pi}{6},\ - \frac{5\pi}{6}$