I just learned about the Harmonic Addition Theorem which applies to functions of the form: $$a\cos x+b\sin x = \operatorname{sgn}(a)\sqrt{a^2+b^2}\;\cos\left(x-\arctan\frac{b}{a}\right) $$
My question is:
Is there a way of generalizing the Theorem to $$a\cos nx + b\sin mx = \text{???}$$
I know, if $n=m$, then
$$a\cos nx +b\sin mx = \operatorname{sgn}(a)\sqrt{a^2+b^2}\;\cos\left(nx-\arctan\frac{b}{a}\right)$$
but that's common sense. I want to know when $n$ and $m$ are not equal. Thanks!