The complex numbers $\sin x+i\cos 2x$ and $\cos x-i\sin 2x$  are conjugate to each other, for

Since, $\left(\overline{\sin x+i\cos 2x}\right)=\cos x-i\sin 2x$

$\Rightarrow \sin x-i\cos 2x=\cos x-i\sin 2x$

$\Rightarrow \sin x=\cos xand\cos 2x=\sin 2x$

$\Rightarrow \tan x=1and\tan 2x=1$

$\Rightarrow x=\pi /4andx=\pi /8$ which is not possible at same time.

Hence, no solution exists.