Well known are the Fourier expansions (presented, e.g., in Abramovitz and Stegun): $$ \cos ( A \sin x) = J_0(A) + 2 \sum_{k=1}^{\infty} J_{2k}(A)~\cos(2kx)~~, $$ $$ \sin ( A \sin x) = 2 \sum_{k=0}^{\infty} J_{2k+1}(A)~\sin((2k+1)x)~~, $$ where $\,J_v\,$ are the Bessel functions.
Similar formulae exist for $ ~\cos ( A \cos x) ~$ and $~ \sin ( A \cos x)~$.
How to generalise them?
Ideally, it would be nice to expand functions like $ \cos ( \sum_i A_i \sin x_i)~$ or, as a starting point, to expand $\cos ( A \sin x + B \sin y)~$. In the latter, we may assume that $x$ and $y$ are commensurate, if that helps.
The simplest case needed is: $$ \cos (~A \sin x + B \sin (x/3)~)~. $$ What would its Fourier expansion look?
We can, of course, employ the afore-presented expansions, but then we shall have to multiply them by one another. I hope there exist a more direct and elegant solution.