Confused by a step in a solution to the problem

Question

I'm pretty confused by the step $$ \prod_{n=1}^{45}\sin(2n^\circ)=\sum_{n=1}^{45}\frac{\omega^n-1}{2i\omega^{n/2}} $$

in the official solution of this problem from 2010 PUMaC Algebra A7:

The expression $\sin2^\circ\sin4^\circ\sin6^\circ\cdots\sin90^\circ$ is equal to $p\sqrt{5}/2^{50}$, where $p$ is an integer. Find $p$.

Can anyone explain this step?

Answer 1

The solutions are publicly available at this link. I can put them in MathJax if you really want, but the link is way easier.