<s>[INST] Fourier Coefficients for $\frac{3}{5-4\cos{2\theta}}$ I want to compute the (even) fourier coefficients for 
 $\dfrac 3 {5-4\cos 2\theta}$ on the interval $[0, 2\pi]$.
Namely, I want to compute the integral:
$$b_n = \int_0^{2\pi} \cos(n\theta) \frac 3 {5-4\cos 2\theta } \frac{d\theta}{2\pi}$$
Integrating this in mathematica, it seems like $b_n = 0$ if $n$ is odd and $b_n = \left(\frac 1 2 \right)^{n/2}$ for $n$ even. 
 [/INST] Hint: Observe
\begin{align}
\frac{3}{5-4\cos 2\theta} = \frac{3/5}{1-\frac{4}{5}\cos 2\theta} = \frac{3}{5}\sum^\infty_{n=0} \left(\frac{4}{5}\right)^n \cos^n 2\theta
\end{align}
and
\begin{align}
\cos^n 2\theta = \left(\frac{e^{2i\theta}+e^{-2i\theta}}{2}\right)^n = \frac{1}{2^n}\sum^n_{k=0}\binom{n}{k} e^{2ik\theta}e^{-2i(n-k)\theta}.
\end{align}
Thus
\begin{align}
\int^{2\pi}_0 e^{2ik\theta}e^{-2i(n-k)\theta} \frac{e^{in\theta}+e^{-in\theta}}{2}\ dx = \text{ something }.
\end{align}
Of course, you could have just consider the integral
\begin{align}
\int^{2\pi}_0 \cos(n\theta) \cos^{n} 2\theta\ \frac{d\theta}{2\pi}.
\end{align}
</s>