Evaluating $\cos\frac{k\pi}{p}+\cos\frac{2k\pi}{p}+\cos\frac{3k\pi}{p}+\dots +\cos\frac{2\frac{(p - 1)}{2}k\pi}{p}$

Question

If $p$ is a prime then what is the value of the series $$\cos\frac{2\pi}{p}+\cos\frac{4\pi}{p}+\cos\frac{6\pi}{p}+\dots +\cos\frac{(p - 1)\pi}{p}$$ In general what is the value of the following series $$\cos\frac{k\pi}{p}+\cos\frac{2k\pi}{p}+\cos\frac{3k\pi}{p}+\dots +\cos\frac{2\frac{(p - 1)}{2}k\pi}{p}$$

Answer 1

In general, if $n$ is an odd positive integer,

$$\sum_{k=0}^{(n-1)/2}\cos(2\pi k/n)=\Re \sum_{k=0}^{(n-1)/2}\exp(2\pi i k/n) =\Re\frac{e^{2\pi i (n-1)/2n}-1}{e^{2\pi i/n}-1} =\Re\frac{-e^{\pi i/n}-1}{e^{2\pi i/n}-1} =\frac12$$

so

$$\sum_{k=1}^{n-1}\cos(2\pi k/n)=\frac12-1=\frac12.$$

The second sum can be handled similarly.