I'm trying to find the Fourier series of the following function:
$$\dfrac{x \cos mx}{1-a\cos x},\ |a|<1.$$
I have discovered the following identity and tried to use it to solve the problem, but had no luck in getting the result. $$\dfrac{1-a\cos x}{1-2a\cos bx+a^2}=1+\sum_{n=1}^{\infty}a^n\cos nbx, |a|<1$$ Is it the right way to calculate this? I find it difficult to transform the numerator to get it closer to the fraction denominator in the left part of the identity; moreover, the fraction will still be inverted, so I won't be able to get the series representation right away.