Let tan - 1 x ∈ - π 2 ,   π 2 , for x ∈ ℝ . Then the number of real…

Given,

$\sqrt{1+\cos 2x}=\sqrt{2}{\tan }^{-1}\left(\tan x\right)$

$\Rightarrow \sqrt{2{\cos }^{2}x}=\sqrt{2}{\tan }^{-1}\left(\tan x\right)$

$\Rightarrow \left|\cos x\right|={\tan }^{-1}\left(\tan x\right)$

Now plotting the diagram of $\left|\cos x\right| \& {\tan }^{-1}\left(\tan x\right)$ we get,

Now from above diagram we can say that,

Number of solutions $=$ Number of intersection points $=3$