I have been studying Backlund transformations using Rogers and Schief, and I am now reading about the permutability theorem. I understood everything up to the very last part for the permutability. It only requires trigonometric identities, but I can't seem to show it myself.
Basically, I need to show that the condition $$\beta_1 \left(\sin\left(\frac{\omega_1 + \omega}{2}\right) - \sin\left(\frac{\Omega + \omega_2}{2}\right)\right) + \beta_2 \left(\sin\left(\frac{\Omega + \omega_1}{2}\right) - \sin\left(\frac{\omega_2 + \omega}{2}\right)\right) = 0$$
is equivalent to
$$\tan\left(\frac{\Omega - \omega}{4}\right) = \frac{\beta_2 + \beta_1}{\beta_2 - \beta_1}\tan\left(\frac{\omega_2 - \omega_1}{4}\right).$$
I have rewritten the second equation as $$\beta_1\left(\tan\left(\frac{\omega_2 - \omega_1}{4}\right) + \tan\left(\frac{\Omega - \omega}{4}\right)\right) + \beta_2\left(\tan\left(\frac{\omega_2 - \omega_1}{4}\right) - \tan\left(\frac{\Omega - \omega}{4}\right)\right) = 0$$ and tried to show part by part. But I can't seem to find that magical step.