How to prove that $\Sigma_{n=1}^{N-1}{\sin^2\frac{p\pi{}n}{N}} = \frac{N}{2}$

Question

Here $p$ is a known integer constant ($p > 0$).
I know that this is true for a fact (checked numerically in Matlab and it holds), but I'm just not able to prove it. I noticed a similar problem related to the sum of reciprocal of sine-squared (Sum of the reciprocal of sine squared), but I'm getting nowhere with this problem.

Answer 1

I can prove that the sum is equal to $N/2$. Here, just recognize that

$$\sin^2{y} = -\frac14 (e^{i y}-e^{-i y})^2 = \frac12 - \frac14 (e^{i 2 y} + e^{-i 2 y}) = \frac12 - \frac12 \Re{[e^{i 2 y}]}$$

so the sum is

$$\begin{align}&= \frac12(N-1) - \frac12 \Re{\sum_{n=1}^{N-1} e^{i 2 p \pi n/N}}\\ &= \frac12(N-1) - \frac12 \Re{\frac{e^{i 2 \pi p/N}-e^{i 2 \pi p}}{1-e^{i 2 \pi p/N}}}\end{align}$$

That last expression is $-1$ because $p \in \mathbb{Z}$. The result follows.

Answer 2

Use the fact $\sin(x) = \frac{1}{2i} [e^{xi}-e^{-xi}]$ (by Euler's formula), expand, and use geometric series.

Answer 3

HINT:

Use $\displaystyle\cos2A=1-2\sin^2A\iff\sin^2A=\frac{1-\cos2A}2$

Then utilize

How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?

or

Sines and cosines of angles in arithmetic progression