n vectors of equal magnitude obey regular polygon law. One of the vectors is along X-axis. Find the value of $\sin \frac{2\pi }{n}+\sin \frac{4\pi }{n}+\dots +\frac{\sin (2n-2)\pi }{n}$ 

Taking component along X-axis $R_X$

$a\sin 0+a\sin \frac{2\pi }{n}+a\sin \frac{4\pi }{n}+\dots .+a\sin \frac{(2n-2)\pi }{n}$

$\text{ But }{\mathrm{R}}_{\mathrm{x}}=0\therefore a\sin 0+a\sin \frac{2\pi }{\mathrm{n}}+\mathrm{asin}\frac{4\pi }{\mathrm{n}}+$

$\dots +a\sin \frac{(2n-2)\pi }{n}=0$

$\therefore a\sin 0+a\sin \frac{2\pi }{n}+a\sin \frac{4\pi }{n}+\dots .+\frac{\sin (2n-2)\pi }{n}=0$.