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

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

Toolbox:
• $1+cos2x=2cos^2x$
Ans - A, no real solution.

$\sqrt{1+cos2x} = \sqrt {2 cos^2x}=\sqrt2\:cosx$
Substituting the value in the given eqn.
$\sqrt{1+cos2x}=\sqrt2\:cos^{-1}cosx$

$\Rightarrow\: \sqrt2\:cosx =\sqrt2\: x$
$\Rightarrow\:cosx=x,$which does not exist for any x.

edited Mar 16, 2013