Maclaurin series for $f(x) = \frac{x-cosa}{1 - 2xcosa + x^{2}}$

Question

I'm trying to compute the Taylor series at 0 for the function $\frac{x-cosa}{1 - 2xcosa + x^{2}}$. The answer in my textbook is: $-\sum_{n=0}^{\infty}cos((n+1)a)x^{n}$.

Although I can calculate it up to the 2nd or 3rd term, I'm having troubles proving the general formula.

I've tried to make a substitution $t = x-cosa$ and find Taylor series at $t=-cosa$ for $\frac{t}{sin^{2}a + t^2}$, but it didn't make it any easier.

Answer 1

Hint: Show that

$$\frac{x-\cos a}{x^2-2x\cos a+1}=\frac{1}{2}\Big(\frac{1}{x-e^{ia}}+\frac{1}{x-e^{-ia}}\Big)$$

Expand around $x=0$:

$$\frac{1}{x-z}=\frac{1}{z}\sum_{n=0}^{\infty}\Big(\frac{x}{z}\Big)^n$$

and simplify for the result.