Explanation

(d)
Given numbers are conjugate to each other,
$\therefore\ \sin{x + i\cos{2x = \cos{x - i\sin{2x}}}}$
$\sin{x = \cos x}$
And $\cos{2x = \sin{2x}}$
$\therefore\ \ \tan{x = 1\ \ \ \Rightarrow \ \ \ \ \ x = \frac{\pi}{4},\ \frac{5\pi}{4},\frac{9\pi}{4},\ldots}$ ...(i)
And $\tan{2x = 1\ \ \ \Rightarrow \ \ \ \ \ 2x = \frac{\pi}{4},\frac{5\pi}{4},\frac{9\pi}{4},\ldots}$
$\Rightarrow \ \ \ \ x = \frac{\pi}{8},\frac{5\pi}{8},\frac{9\pi}{8},\ldots$ ...(ii)
There exists no value of $x$ common in Eqs. (i) and (ii)