Graph?

$RHS:5^x+\dfrac{1}{5^x}\geq 2~~(\because\color{#D61F06}{AM\geq GM})$

$LHS: \sin(e^x)\in[-1,1]\forall x\in\mathbb R$

Hence there is no $x\in \mathbb{R}$ for which equation is true. Hence answer is $\boxed{0}$ .

$-1 \le \sin(e^{x}) \le 1$
Using AM-GM inequality,

$5^{x} + 5^{-x} \ge 2$
Thus there exists no solution.