Here’s a problem I started working on today:Express $\sin^{2n}(\theta)$ as a sum of sines whose angles are multiples of $\theta$.
As a hint we have $e^{in\theta}-e^{-in\theta}=2i\sin(n\theta)$
I can’t seem to figure out how to show this. If it were cosine, this would have been a different story – just pairing terms. But I can’t even figure out how to express $\sin^2(\theta)$ as a sum of sines. Any hints?