What is $\cos\frac{\pi}{13}+\cos\frac{3\pi}{13}+\cos\frac{5\pi}{13}+\cos\frac{7\pi}{13}+\cos\frac{9\pi}{13}+\cos\frac{11\pi}{13}$?
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?