Sum of trigonometric secants

Question

I don't know what formula to use to solve the next sum

$\textrm{sec}\left ( \frac{2\pi }{7} \right )+\textrm{sec}\left ( \frac{4\pi }{7} \right )+\textrm{sec}\left ( \frac{6\pi }{7} \right )$

Please, give me some advice. Thanks for your help.

Answer 1

$$\textrm{sec}\left ( \frac{2\pi }{7} \right )+\textrm{sec}\left ( \frac{4\pi }{7} \right )+\textrm{sec}\left ( \frac{6\pi }{7} \right )=\frac{\cos\frac{2\pi}{7}\cos\frac{4\pi}{7}+\cos\frac{2\pi}{7}\cos\frac{6\pi}{7}+\cos\frac{4\pi}{7}\cos\frac{6\pi}{7}}{\cos\frac{2\pi}{7}\cos\frac{4\pi}{7}\cos\frac{6\pi}{7}}=$$ $$=\frac{\cos\frac{6\pi}{7}+\cos\frac{2\pi}{7}+\cos\frac{6\pi}{7}+\cos\frac{4\pi}{7}+\cos\frac{4\pi}{7}+\cos\frac{2\pi}{7}}{2\cos\frac{2\pi}{7}\cos\frac{4\pi}{7}\cos\frac{8\pi}{7}}=$$ $$=\frac{\cos\frac{2\pi}{7}+\cos\frac{4\pi}{7}+\cos\frac{6\pi}{7}}{\cos\frac{2\pi}{7}\cos\frac{4\pi}{7}\cos\frac{8\pi}{7}}=$$ $$=\frac{\frac{2\sin\frac{\pi}{7}\cos\frac{2\pi}{7}+2\sin\frac{\pi}{7}\cos\frac{4\pi}{7}+2\sin\frac{\pi}{7}\cos\frac{6\pi}{7}}{2\sin\frac{\pi}{7}}}{\frac{8\sin\frac{2\pi}{7}\cos\frac{2\pi}{7}\cos\frac{4\pi}{7}\cos\frac{8\pi}{7}}{8\sin\frac{2\pi}{7}}}=$$ $$=\frac{\frac{\sin\frac{3\pi}{7}-\sin\frac{\pi}{7}+\sin\frac{5\pi}{7}-\sin\frac{3\pi}{7}+\sin\frac{7\pi}{7}-\sin\frac{5\pi}{7}}{2\sin\frac{\pi}{7}}}{\frac{\sin\frac{16\pi}{7}}{8\sin\frac{2\pi}{7}}}=\frac{-\frac{1}{2}}{\frac{1}{8}}=-4.$$