Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2004 / 5 / ii / Solution

For a nonnegative integer $n$, put $z=e^{i\theta }$ on the unit circle. Since $\cos \theta =(z+z^{-1})/2$ and $d\theta =dz/(iz)$,The residue at zero is the coefficient of $z^{2n}$ in $(1+z^2{)}^{2n}$, namely $\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$. The residue theorem givesThe contour integrand has no other poles. The formula also includes $n=0$, giving $2\pi$.