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

We know that

$\begin{array}{l}\mathrm{AM}\ge \mathrm{GM}\\ \therefore \frac{2^x+2^{-x}}{2}\ge \sqrt{2^x\times 2^{-x}}\\ \Rightarrow 2^x+2^{-x}\ge 2\end{array}$

But, sin  $\left(e^x\right)\le 1$

Thus, $\sin \left(e^x\right)\ne 2^x+2^{-x}$ for any $x\in R$

Hence, the given equation has no solution.