I found this problem recently on Yahoo Answers! It is gone now because it went unanswered after four days, and it is the policy of YA! to delete such questions.

Given:

$\displaystyle {I(n)=\int^{\pi/2}_{-\pi/2}\frac{\cos(2nx)}{(1+e^x)\cos(2x)}dx}$

Prove:

$\displaystyle I(n + 2) = -I(n)$, for all natural numbers $\displaystyle n$. (Since "natural number" is ambiguous, let's say for non negative—or possibly positive integers.)

I couldn't get passed the singularities at $\displaystyle -\pi/4$ and $\displaystyle \pi/4$ . WolframAlpha couldn't provide an answer either, even for (e.g.), $\displaystyle n=2$.

I have my doubts about this being true; however, if it is I would like to see a proof.