Problem 12.4. 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 $\alpha \neq \frac{k \pi}{2}, \quad k \in Z$.

Emil Kolev
12.4. The equality is equivalent to

$$
\sin (n-1) \alpha=\frac{(n-1) \sin 2 \alpha}{2}
$$

When $n \geq 4$ setting $\alpha=\frac{\pi}{4}$ gives

$$
\sin \left((n-1) \frac{\pi}{4}\right)=\frac{n-1}{2} \geq \frac{3}{2}
$$

a contradiction.

When $n=1$ and $n=3$ the equality (1) is an identity and when $n=2$ we have $\sin \alpha=\frac{\sin (2 \alpha)}{2}$, which is not true for $\alpha=\frac{\pi}{4}$. Therefore the answer is $n=1$ and $n=3$.