so Im stuck on the following problem;
Use the identity
$\exp(ix\sin\theta) = \sum\limits_{k=-\infty}^\infty J_k(x)\exp(ik\theta)$
to find the Fourier series of $\cos(\theta + 4\sin\theta)$, where $J_k$ is the $k$th Bessel functions.
I tried to expand $\cos(\theta + 4\sin\theta)$ using trig identities and use it to find $A_n$ and $B_n$ the usual way for Fourier series, but this lead to an ugly integral that maple cant even solve. Where does the identity fit in? Even anyone has any hints to start it would be greatly appreciated!
Update: continue using Mhenni's hint
Try to compute $A_n$:
$\cos(\theta+4\sin t) = (1/2)(\exp(i(t+4\sin t))+\exp(-i(t+4\sin t) $
$= (1/2)(e^{it}\sum J_k(4)e^{ikt} + e^{-it}\sum J_k(-4)e^{ikt})$
$A_n = (1/2\pi)\int\limits_{-\pi}^{\pi}\sum(J_k(4)e^{it+ikt}+J_k(-4)e^{-it+ikt})\cos(mt)dt$
is this looking any better? or have I gone wrong anywhere? as when I integrate this using maple it looks pretty bad too.