How to prove an Identity

Question

I'm reading Pogorelov's Analytic geometry and in page 94 he states the following: "Let, $A_1,A_2,...,A_n$ be the vertices of a regular n-gon. Then $\vec x_1+\vec x_2+...+\vec x_n=0$ where $\vec x_1,\vec x_2,...,\vec x_n$ are vectors directed from each vertex to the other going round the n-gon in either direction". Derive from this that: $$1+\cos \frac{2\pi}{n}+\cos \frac{4\pi}{n}+...+\cos \frac{(2n-2)\pi}{n}=0$$ $$\sin \frac{2\pi}{n}+\sin \frac{4\pi}{n}+...+\sin \frac{(2n-2)\pi}{n}=0$$ I didn't have issues confirming the sum of the vectors, in the other hand I can't prove the last two identities so I would gladly accept any help you can provide.