solve a complex integral

Question

I stumbled on this integral, the problem says to solve it with contour integration. Any insights on how to solve this in function of $n$?

\begin{equation} \int_{0}^{2\pi}\cos^{2n}(\theta)d\theta \end{equation}

Answer 1

$$\cos\theta = \frac{e^{i\theta}+e^{-i\theta}}{2}.$$ Set $z=e^{i\theta}$ and compute the residue in the origin of $g(z)=\frac{1}{z}\left(\frac{z+1/z}{2}\right)^{2n}.$