The number of real solutions of the equation$\sqrt{1+\cos2x}=\sqrt 2\cos^{-1}(\cos x)\;in\begin{bmatrix}\frac{\Large \pi}{2},\pi\end{bmatrix}$

Question

$$(A)\quad 0\quad(B)\quad 1\quad(C)\quad 2\quad(D)\quad \text{Infinite}$$