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

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

Then:
 * $\displaystyle \int_0^{\frac \pi 2} \sin^{2 n} x \rd x = \dfrac {\paren {2 n}!} {\paren {2^n n!}^2} \dfrac \pi 2$