Evaluating $\sum \sin(ak) x^k / k! $

Question

Mathematica tells me that $$ \sum_{k=0}^\infty \sin(ak) \frac{x^k}{k!} = e^{x \cos (a)} \sin (x \sin (a)). $$ How would I go about evaluating such a series by hand? My first thought is to expand $\sin(ak)$, obtaining $$ \sum_{k=0}^\infty \sum_{n=0}^\infty \frac{(-1)^n (ak)^{2n+1} x^k}{k! (2 n+1)!} = \sum_{n=0}^\infty \frac{(-1)^n a^{2n+1}}{(2 n+1)!} \sum_{k=0}^\infty \frac{k^{2n+1} x^k}{k!} = e^x \sum_{n=0}^\infty \frac{(-1)^n a^{2n+1}}{(2 n+1)!} B_{2n+1}(x)$$ where $B_n(x)$ is a Bell polynomial. However, it is unclear to me how to proceed from here.

Answer 1

$$ \sum_{k=0}^\infty \sin(ak) \frac{x^k}{k!} = \Im \left[\sum_{k=0}^\infty e^{iak} \frac{x^k}{k!}\right]\\ = \Im [e^{xe^{ia}}] = e^{x \cos (a)} \sin (x \sin (a)). $$