Why does the following identity hold?
$$\int\_0^{2\pi}\int\_0^{2\pi}\arcsin(k\cdot \cos(\theta-\phi))e^{i(\theta-\phi)} \, d\theta \, d\phi=2\pi\int\_0^{2\pi}\arcsin(k\cdot \cos x)e^{ix} \, dx$$
where $k\in[-1,1]$ is a constant.

I'm trying to use substitution $x=\theta-\phi$ but it still needs some tricks. Thanks!