Cosine is dancing

$S = \cos \dfrac{2\pi}{2013}+\cos \dfrac{4\pi}{2013}+\cos \dfrac{6\pi}{2013}+...+\cos \dfrac{2010\pi}{2013}+\cos \dfrac{2012\pi}{2013}$

$2S \sin \frac{\pi}{2013} = 2 \sin \frac{\pi}{2013}\cos \dfrac{2\pi}{2013}+2 \sin \frac{\pi}{2013}\cos \dfrac{4\pi}{2013}+2 \sin \frac{\pi}{2013}\cos \dfrac{6\pi}{2013}+...+2 \sin \frac{\pi}{2013}\cos \dfrac{2010\pi}{2013}+2 \sin \frac{\pi}{2013}\cos \dfrac{2012\pi}{2013}$

$2S \sin \frac{\pi}{2013} = [\sin \frac{3\pi}{2013} - \sin \frac{\pi}{2013}] + [\sin \frac{5\pi}{2013} - \sin \frac{3\pi}{2013}] + ... + [\sin \frac{2013\pi}{2013} - \sin \frac{2012\pi}{2013}]$

$2S \sin \frac{\pi}{2013} = \sin \frac{2013\pi}{2013}- \sin \frac{\pi}{2013} = - \sin \frac{\pi}{2013}$

$S = - \frac{1}{2}$

Another solution:

Consider the equation $x^{2013}=1$

$\implies 1+\sum_{k=1}^{2012} \cos\dfrac{2k\pi}{2013} =0$

$1+2\sum_{k=1}^{1006} \cos\dfrac{2k\pi}{2013} =0$

So $\cos \dfrac{2\pi}{2013}+\cos \dfrac{4\pi}{2013}+\cos \dfrac{6\pi}{2013}+...+\cos \dfrac{2010\pi}{2013}+\cos \dfrac{2012\pi}{2013}= -0.5$