The number of real solutions of the equation 1+ 2 x = 2 ^ -1 ( x)...

\begin{aligned} & & \sqrt{1+\cos 2 x} & =\sqrt{2} \sin ^{-1}(\sin x) \\ \Rightarrow & & \sqrt{2}|\cos x| & =\sqrt{2} \sin ^{-1}(\sin x) \\ \Rightarrow & & |\cos x| & =\sin ^{-1}(\sin x) \end{aligned} When we draw the graph both functions (shown below) we can actually see that they intersect only at two points $\forall x \in-\pi \leq x \leq \pi$.