The number of real solutions of the equation $\cos \left(e^x\right)=2^x+2^{-x}$  is:

       $\cos \left(e^x\right)=2^x+2^{-x}$                             …..(i)

$\because$       $AM\ge GM$

$\Rightarrow$       $\frac{2^x+2^{-x}}{2}\ge \sqrt{2^x{.2}^{-x}}$

$\Rightarrow$         $2^x+2^{-x}\ge 2$

$\Rightarrow$        $\cos \left(e^x\right)\ge 2$                         [from Eq. (i)]

impossible$\left(\because -1\le \cos \theta \le 1\right)$,

$\Rightarrow$ no solution ie; zero number of solutions.