$\cos \alpha+\cos(\alpha+\beta)+\cos(\alpha+2\beta)+.....+\cos(\alpha+(n-1)\beta)=0 $ If each side of a regular polygon of $n$ sides subtend an angle $\alpha$ at the center of the polygon and each exterior angle of the polygon is $\beta$,then prove that $\cos \alpha+\cos(\alpha+\beta)+\cos(\alpha+2\beta)+.....+\cos(\alpha+(n-1)\beta)=0 $
Since this is a regular polygon.Therefore,each $\alpha=\frac{2\pi}{n}$ and since each external angle is $\beta$.So by geometry,$\alpha=\beta.$
$\cos \alpha+\cos(\alpha+\beta)+\cos(\alpha+2\beta)+.....+\cos(\alpha+(n-1)\beta)=\frac{\cos\frac{n\beta}{2}}{\cos\frac{\beta}{2}}\cos\frac{2\alpha+(n-1)\beta}{2}$
Now putting $\alpha=\beta=\frac{2\pi}{n}$ does not give me answer.What mistake did i make?

Notice, in a regular polygon with $n$ number of side, 
the angle subtended by each side at the center of the polygon $$\alpha=\frac{2\pi}{n}$$ 
each exterior angle of the polygon $$\beta=\pi-\frac{(n-2)\pi}{n}=\frac{2\pi}{n}$$ 
$$\implies \alpha=\beta=\frac{2\pi}{n}$$  Now, we have 
$$\cos \alpha+\cos(\alpha+\beta)+\cos(\alpha+2\beta)+\ldots +\cos(\alpha+(n-1)\beta)$$ setting $\beta=\alpha$
$$=\cos \alpha+\cos(\alpha+\alpha)+\cos(\alpha+2\alpha)+\ldots +\cos(\alpha+(n-1)\alpha)$$
$$=\cos \alpha+\cos(2\alpha)+\cos(3\alpha)+\ldots +\cos(n\alpha)$$ $$=\frac{1}{2}\left[\sin n\alpha\cot\frac{\alpha}{2}+\cos n\alpha-1\right]$$
Setting $\alpha=\frac{2\pi}{n}$ $$=\frac{1}{2}\left[\sin n\left(\frac{2\pi}{n}\right)\cot\frac{1}{2}\left(\frac{2\pi}{n}\right)+\cos n\left(\frac{2\pi}{n}\right)-1\right]$$
$$=\frac{1}{2}\left[\sin 2\pi\cot\frac{\pi}{n}+\cos 2\pi-1\right]$$
$$=\frac{1}{2}\left[0+1-1\right]=0$$ Hence, proved that
$$\bbox[5px, border:2px solid #C0A000]{\color{red}{\cos \alpha+\cos(\alpha+\beta)+\cos(\alpha+2\beta)+\ldots +\cos(\alpha+(n-1)\beta)=\color{blue}{0}}}$$
