Question 112
Find all positive integers $n$ for which the equality
$$
\frac{\sin (n \alpha)}{\sin \alpha}-\frac{\cos (n \alpha)}{\cos \alpha}=n-1
$$
holds true for all $\alpha \neq \frac{k \pi}{2}, \quad k \in \mathbb{Z}$.
$$
\frac{\sin (n \alpha)}{\sin \alpha}-\frac{\cos (n \alpha)}{\cos \alpha}=n-1
$$
holds true for all $\alpha \neq \frac{k \pi}{2}, \quad k \in \mathbb{Z}$.