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

$\begin{array}{l}Conjugateof \sin x+icon2xis\sin x-i\cos 2x\Rightarrow \sin x-i\cos 2x=\cos x-i\sin 2x\\ \Rightarrow \sin x=\cos x,\cos 2x=\sin 2x\Rightarrow \tan x=1and\tan 2x=1\end{array}$

$\Rightarrow x$has no real value.