Simple Trignometric Equation

Question

If $ N $ is some positive integer and $ \alpha $ some real number, how do I solve $$ \frac{\sin( x (N+1) )}{\sin(x N )} = \alpha $$ for x? Or is there no simple closed form expression for x?

Thank you!

Answer 1

$\sin(nx)$ and $\cos(nx)$ are always n-th order polynomials in $\sin x$ and/or $\cos x$, so there's that, but, then again, not even polynomials can be solved analytically for $n>4.$