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

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

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