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

sin x + i cos 2x and cos x – i sin 2x are conjugate of each other

$\begin{array}{l}\Rightarrow \sin \mathrm{x}+\mathrm{icos}2\mathrm{x}=\cos \mathrm{x}+\mathrm{isin}2\mathrm{x}\\ \Rightarrow \sin \mathrm{x}=\cos \mathrm{x}\text{ and }\cos 2\mathrm{x}=\sin 2\mathrm{x}\\ \Rightarrow \tan \mathrm{x}=1\text{ and }\tan 2\mathrm{x}=1\end{array}$

which is not possible for any value of x.