Is there generally a way to turn a (finite) Fourier series:
$$ f(x) := a_0 + \sum_{n=1}^{N-1} a_k \cos(nx) + b_k \sin(nx) $$
Into a product? eg:
$$ 1 - \cos(2x) + \cos(4x) = \frac{\cos(2x) \cos(3x)}{\cos(x)} $$
I can generally find the series representation of a product like this by evaluating it at a few points and taking the discrete fourier transform. I don't know if all such products have a finite series representation, but the small handful I've tried do.
However, I don't know how to go from the series representation to the corresponding product, or even if all series have a corresponding product.