Hunter become the huntee

Calculus Level 4

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

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 \(n\).

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