Definite Integral from 0 to Half Pi of Even Power of Cosine x

Theorem
Let $n \in \Z_{> 0}$ be a positive integer.

Then:
 * $\displaystyle \int_0^{\frac \pi 2} \cos^{2 n} x \rd x = \dfrac {\left({2 n}\right)!} {\left({2^n n!}\right)^2} \dfrac \pi 2$