Find all positive integer $n$ such that
$$\frac{\sin{n\theta}}{\sin{\theta}} - \frac{\cos{n\theta}}{\cos{\theta}} = n-1$$holds for all $\theta$ which are not integral multiples of $\frac{\pi}{2}$
We have $\frac{sin n\theta cos\theta -cos n\theta sin \theta}{sin\theta cos\theta} =n-1$
Which is $\frac{sin((n-1)\theta)}{sin\theta cos\theta}=n-1$
So $sin((n-1)\theta )=\frac{n-1}{2}sin 2\theta$
Let $\theta =\frac{\pi}{4}$
We get $sin ((n-1)\frac{\pi}{4})=\frac{n-1}{2}$
So $\frac{n-1}{2} \in \{ 1,\frac{\sqrt{2}}{2},0,-\frac{\sqrt{2}}{2},-1 \}$
Give $n=1$ or $3$