$\cos \frac{2\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{6\pi }{7} =$

$\begin{array}{rcl} & & \cos \frac{2\mathrm{\pi }}{7}+\cos \frac{4\mathrm{\pi }}{7}+\cos \frac{6\mathrm{\pi }}{7}\\ & =& \frac{2\sin {\displaystyle \frac{2\mathrm{\pi }}{7}} \cos {\displaystyle \frac{2\mathrm{\pi }}{7}}+2\cos {\displaystyle \frac{4\mathrm{\pi }}{7}} \sin {\displaystyle \frac{2\mathrm{\pi }}{7}}+2\cos {\displaystyle \frac{6\mathrm{\pi }}{7}} \sin {\displaystyle \frac{2\mathrm{\pi }}{7}}}{2\sin {\displaystyle \frac{2\mathrm{\pi }}{7}}}\\ & =& \frac{\sin {\displaystyle \frac{4\mathrm{\pi }}{7}}+\sin {\displaystyle \frac{6\mathrm{\pi }}{7}}-\sin {\displaystyle \frac{2\mathrm{\pi }}{7}}+\sin {\displaystyle \frac{8\mathrm{\pi }}{7}}}{2\sin {\displaystyle \frac{2\mathrm{\pi }}{7}}}\\ & =& \frac{\sin \left(\mathrm{\pi }-{\displaystyle \frac{\mathrm{\pi }}{7}}\right)-\sin {\displaystyle \frac{2\mathrm{\pi }}{7}}+\sin \left(\mathrm{\pi }+{\displaystyle \frac{\mathrm{\pi }}{7}}\right)}{2\sin {\displaystyle \frac{2\mathrm{\pi }}{7}}}\\ & =& \frac{\sin {\displaystyle \frac{\mathrm{\pi }}{7}}-\sin {\displaystyle \frac{2\mathrm{\pi }}{7}}-\sin {\displaystyle \frac{\mathrm{\pi }}{7}}}{2\sin {\displaystyle \frac{2\mathrm{\pi }}{7}}}\\ & =& \frac{-\sin {\displaystyle \frac{2\mathrm{\pi }}{7}}}{2\sin {\displaystyle \frac{2\mathrm{\pi }}{7}}}=-\frac{1}{2}\end{array}$