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Jul 1, 2013 at 14:18 comment added David E Speyer I'm not sure what counts as elementary in this game, but I would think the easiest proof that $\frac{1}{\pi} \int \cos^{2n} x dx = \binom{2n}{n}/4^n$ is to write $\cos x = (e^{ix} + e^{-ix})/2$ and expand $\cos^{2n}$ by the binomial theorem.
Jun 16, 2013 at 13:18 vote accept RandomStudent
Jun 16, 2013 at 13:26
Jun 14, 2013 at 14:40 history answered Noam D. Elkies CC BY-SA 3.0