Say you have the equations:
\begin{align} -S_1\sin\left(2\psi+\theta\right)+S_2\cos\left(\psi\right)&=S_3\\ S_1\cos\left(2\psi+\theta\right)+S_2\sin\left(\psi\right)&=S_4 \end{align}
or switching around the phase, which is perfectly ok for the system I am studying:
\begin{align} -S_1\sin\left(2\psi\right)+S_2\cos\left(\psi+\theta\right)&=S_3\\ S_1\cos\left(2\psi\right)+S_2\sin\left(\psi+\theta\right)&=S_4 \end{align}
and wish to eliminate $\psi$. How to do that? I have tried using 'square and add', rewriting using double-angle terms, and using Maple - without any luck so far.