Let us define function $f:[0~ 2\pi] \rightarrow R$ as follows:

\begin{align} f(x)\triangleq \sum_{i=1}^K \frac{\alpha_i \gamma_i \sin(x-\theta_i)}{1+\gamma_i[1+\cos(x-\theta_i) ]}, \end{align} where anything except $x$ is a given parameter and we have $\alpha_i >0, \forall i$ and $\gamma_i >0, \forall i$. I am trying to find the closed form solutions of $f(x)=0$ in interval $[0~2\pi]$. Does anybody know how to do that? If it is not possible to find a closed form solution, can we say something about the number of solutions? Is there finite number of solutions?

Any help would be appreciated.