I need to solve:

$\displaystyle \int_{-\pi \over 2}^{\pi \over 2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}(x)}{\sin^{2008}(x)+\cos^{2008}(x )}dx$

This integral is beyond my capabilities! This was given to me as challenge by my classmate. According to him the answer is $\displaystyle \pi \over 4$.

What should I do to solve it?