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Let $m \in \mathbb{N}: m > 2$, and define $\theta_{i} = \frac{2\pi*(i-1)}{m} \forall i \leq m$. How can I show that $\sum_{i=1}^{m}(cos(2 \theta_{i})) = \sum_{i=1}^{m}(cos(\frac{4\pi(i-1)}{m})) =0$ by means of complex numbers and geometric sequences?

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up vote 0 down vote accepted

$cos(2 \theta_{i})$ is the real part of a root of unity. Add up the real parts of each corresponding root of unity; they cancel perfectly to 0.

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