Extended Trigonometric Pythagorean Identities: The Proof

I recently posted on Extended Trigonometric Pythagorean Identities and a “supercharged” version of them, which allows to simplify certain sums of shifted sine functions raised to integer powers. In particular, the claim was that $$\sum_{n=0}^{N-1} \sin^2\left(x+n\frac{\pi}{N}\right) = \frac{N}{2}$$ or more generally for any integer $k$ and $N\geq k+1$: $$\sum_{n=0}^{N-1} \sin^{2k}\left(x+n\frac{\pi}{N}\right) = N \frac{(2k)!}{(k!)^2 2^{2k}} = […]

Trigonometric Pythagorean Identity, supercharged

You know how they say good things always come back? Well, I recently stumbled over something that reminded me a lot on a post I had made about generalizations of the Trigonometric Pythagorean Identity. In short, the well-known identity $\sin^2(x)+\cos^2(x)=1$ can be generalized to a sum of $N\geq 2$ terms that are uniformly shifted copies […]

Trigonometric Pythagorean Identity, extended

Here is to another curious and funny identity that has popped up many times and I’ve never quite gotten to find a proof for it: $$ \sum_{n=0}^{N-1} \sin^2\left( x+ \frac{\pi}{N} n \right) = \frac{N}{2}, \; \forall N \in \mathbb{N} \geq 2$$ The Trigonometric Pythagorean Identity What’s the connection to good old Pythagoras you ask? Well, […]