Wallis Formula for Definite Integrals of Sine/Cosine Powers

$$\lt br/\gt \int_0^{\pi/2}\sin^n x\,dx=\lt br/\gt \begin{cases}\lt br/\gt \dfrac{(n-1)!!}{n!!}\,\dfrac{\pi}{2},&n\;\mathrm{even},\\[6pt]\lt br/\gt \dfrac{(n-1)!!}{n!!},&n\;\mathrm{odd},\lt br/\gt \end{cases}\lt br/\gt \quad\lt br/\gt \int_0^{\pi/2}\cos^n x\,dx=\int_0^{\pi/2}\sin^n x\,dx.\lt br/\gt $$
where $$n\in\mathbb{N}_0$$ and $$!!$$ denotes the double factorial.