Number of unique solutions to $\sin P_1(x, n_1) = \sin P_2(x,n_2)$

Question

In attempting to answer this question, I was looking at the solutions for $\sin(3x - 4) = \cos(7x)$ when $0 \leq x \leq 2\pi$ (all other solutions should be multiples of these). I found $14$ distinct solutions using Wolfram Alpha, which surprised me greatly. Thinking further, I wondered if one can determine the number of unique solutions (those that are not a factor of $2\pi n, n\in \mathbb{N}$ from another solution in the list) for $\sin P_1(x, n_1) = \sin P_2(x,n_2)$, where $P_1(x, n_1)$ and $P_2(x,n_2)$ are polynomials of degrees $n_1$ and $n_2$ respectively. Does anyone know of a way this could be determined in general without analyzing specifics? Any closed form would do... I just want to be able to quickly check whether or not I have found all the solutions for such a problem while working one.