It is widely known that $ \frac{\sin\alpha+\sin\beta}{\cos\alpha+\cos\beta} = \tan\frac{\alpha+\beta}{2}. $
I'm wondering if the following is "known" in the sense of being in published sources?
Suppose $\alpha,\beta,\gamma\in(-\pi/2,\pi/2)$. $ \text{If }\frac{\sin\alpha\sin\beta}{\cos\alpha+\cos\beta} = \tan\gamma,\text{ then }\tan\frac\gamma2 = \tan\frac\alpha2\cdot\tan\frac\beta2. $