Let $\alpha =\frac{\pi}{n}, n \in \mathbb{N}, n \ge 3$ Prove that: $$\sin\alpha\sin2\alpha + \sin2\alpha\sin3\alpha + \ldots + \sin(n-2)\alpha\sin(n-1)\alpha = \frac{n}{2}\cos\alpha$$
The problem is from a high school trigonometry workbook.
I attempted to use induction and successfully proved the base case; however, I am uncertain about the induction hypothesis because $\alpha$ is also expressed in terms of $n$. Will it work? If I assume that the statement is true for a natural number $n$ and try proving the case for $n + 1$, can I actually use the hypothesis? Or would a different approach be more suitable?
Hope someone can help! Thanks!