I'm currently studying Beta and Gamma functions and there is problem I'm struggling with: I have to write the solution of the following integral in form of a double factorial: $$\int_{0}^{\pi/2}\cos^n(\theta)\, d\theta$$ After doing a substitution and integrating I got: \begin{align} \int_{0}^{\pi/2}\cos^n(\theta)\, d\theta&=\frac{1}{2}B\left(\frac{n+1}{2},\frac{1}{2}\right)\\&=\frac{1}{2}\frac{\Gamma(\frac{n+1}{2})\Gamma(\frac{1}{2})}{\Gamma(\frac{n}{2}+1)}\\ &=\frac{\sqrt{\pi}}{2}\frac{(n+2)}{(n+1)}\frac{(\frac{n+1}{2})!}{(\frac{n}{2}+1)!} \end{align}
We never dealt with a double factorial in our class and I don't know why the solution has to be in this form but I would be really glad if someone could help me out symplifying the solution.
Also any hints on how to transform this to a double factorial would be appreciated, thanks!