The values of x for which $\sin x+i\cos 2x$ and $\cos x-i\sin 2x$ are conjugate to each other are

Conjugate of $\cos x-i\sin 2x$  is  $\cos x+i\sin 2x$

So, $\sin x+i\cos 2x=\cos x+i\sin 2x$ compare

  $\sin x=\cos x$                        $\cos 2x=\sin 2x$

 $\tan x=1$                                $\tan 2x=1$

 But for some value of x, both $\tan x$ and $\tan 2x$ can not be equal.

$\therefore$ No solution