[math] \newcommand{\mathds}{\mathbb}[/math]

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Establish the following formula,

[[math]] \int_0^{\pi/2}\cos^pt\,dt=\int_0^{\pi/2}\sin^pt\,dt=\left(\frac{\pi}{2}\right)^{\varepsilon(p)}\frac{p!!}{(p+1)!!} [[/math]]

where [math]\varepsilon(p)=1[/math] if [math]p[/math] is even, and [math]\varepsilon(p)=0[/math] if [math]p[/math] is odd.