I know the Jacobi-Anger expansion relation which gives the Fourier development of $e^{i z \cos(\theta)}$ and ${ e^{i z \sin(\theta)} }$, such that $$ \begin{cases} e^{i z \cos(\theta)} = \sum\limits_{n} i^{n} \mathcal{J}_{n} (z) \, e^{i n \theta} \, , \\ e^{i z \sin(\theta)} = \sum\limits_{n} \mathcal{J}_{n} (z) \, e^{i n \theta} \, , \end{cases} $$ where ${ \mathcal{J}_{n} }$ are Bessel functions of the first kind.
I am interested in possible generalizations of this relation to higher harmonics. Are there any known results for the Fourier development of $$ e^{i z \cos(2 \theta)} \;\;\; \text{and} \;\;\; e^{i z \sin(2 \theta)} \;\;\; \text{?} $$ More generally, are there know results for $e^{i z \cos(n \theta)}$ and $e^{i z \sin (n \theta)}$, with ${ n \in \mathbb{N}}$ ?
Any enlightning references are also, of course, welcome.