Given this integral
$$\int_{-\infty}^{+\infty}\sin(\cosh x)\cos(\sinh x)\mathrm dx={\pi\over 2}\tag1$$
My try:
Recall $$2\sin(A)\cos(B)=\sin(A-B)+\sin(A+B)$$
$(1)$ becomes
$$\frac12\int_{-\infty}^{+\infty}\cos(\cosh x-\sinh x)+\sin(e^x)\mathrm dx\tag2$$
Recall $$\cosh^2 x-\sinh^2 x=1$$
$(2)$ becomes
$$\frac12\int_{-\infty}^{+\infty}\sin(e^{-x})+\sin(e^x)\mathrm dx\tag3$$
How may be prove $(1)?$