Fourier series of $\frac{1-r^2}{1-2r\cos x+r^2}$ when $|r|<1$

Question

How can I calculate the Fourier series of $\frac{1-r^2}{1-2r\cos x+r^2}$ when $|r|<1$? I know the answer is $1+\sum_{n=1}^{\infty}2r^n\cos nx$, but is there any way to solve the problem in the realm of real numbers? Thank you for your help.