The value of $\sum _{k=1}^{n-1}(n-k)\cos \frac{2k\pi }{n}$       ( $n\ge 3$  is an integer)

$S=(n-1)\cos \frac{2\pi }{n}+(n-2)\cos \frac{4\pi }{n}+(n-3)\cos \frac{6\pi }{n}+\mathrm{......}+\cos \frac{2(n-1)\pi }{n}$$S=1\cos \frac{2\pi }{n}+2\cos \frac{4\pi }{n}+\mathrm{..........}+(n-1)\cos \frac{2(n-1)\pi }{n}$

$2S=n(\cos \frac{2\pi }{n}+\cos \frac{4\pi }{n}+\mathrm{.......}+\cos \frac{2(n-1)\pi }{n})$

$2S=n\frac{\sin (n-1)\frac{\pi }{n}}{\sin \frac{\pi }{n}}\cos \left(\frac{\frac{2\pi }{n}+\frac{2(n-1)\pi }{n}}{2}\right)=-n$