Hunter become the huntee

$\large \displaystyle \int_{0}^{\frac{\pi}{2}}\cos ^{n}\left ( \varphi \right )\, d\varphi$

If the value of the integral above can be expressed as $\displaystyle \frac{2^{2\cdot 864}\left ( 864! \right )^{2}}{\left ( 2\cdot 864+1 \right )!}$ for constant $n$ , find the value of $n$ .