I came across beautiful integral (where $n$ is integer)
$I(n, z) = \int_0^{\pi} \cos(nx) \sin(z \cos(x) ) \mathrm{d}x $
According to Gradshteyn and Ryzhik (p 414, Sec. 3.715, Eq. 13), solution is with Bessel function:
$I(n, z) = \pi \sin(n \pi /2) J_n(z)$
How to proove beautiful result?
For even $n$, just do $x\mapsto x-\pi/2$ and use the fact that your integrand is becoming odd (and you integrate over the symmetric interval $[-\pi/2,\pi/2]$.
For odd $n$, inserting $t=ie^{ix}$ into the generating function expansion $$ e^{\frac{z}{2}(t-1/t)}=\sum_{k=-\infty}^{+\infty} J_k(z) t^k, $$ we find the Fourier cosine series $$ e^{iz\cos x}=J_0(z)+\sum_{k=1}^{+\infty}i^k J_k(z)\cos(kx). $$ Take the imaginary part, and your integral formula is given by the formula for the Fourier coefficient.