I am reading about Wigner matrices right now and have come across the computation for even moments $$\frac{2\cdot 2^k}{\pi}\int_{\pi/-2}^{\pi/2} \sin^{2k}x \cos^2x \; dx = C_k$$ where $C_k = \frac{1}{k+1}\binom{2k}{k}.$ I've considered a couple cases for different values of $k$ to see if I could conclude how one arrives at the Catalan numbers, but I don't see it.
Can anyone explain how this computation is done?