I'd love your help with proving that the following series $\sum_{n=0}^{\infty}\frac{\sin (nx)}{n!}$ is the Fourier series of $e^{\cos x} \sin (\sin x)$.
I tried to find $\hat f(n)$ using integration by parts and I tried to use Taylor's series of $e^x$ in order to get $n!$ by I didn't reach to anything close of what I should.
Thanks a lot.