referenence request: yet another tangent half-angle formula

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 $\alpha,\beta,\gamma\in(-\pi/2,\pi/2)$. $$\text{If }\frac{\sin\alpha\sin\beta}{\cos\alpha+\cos\beta} = \tan\gamma,\text{ then }\tan\frac\gamma2 = \tan\frac\alpha2\cdot\tan\frac\beta2.$$

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Maybe I need to say "If $\dfrac{\sin\alpha\sin\beta}{\cos\alpha+\cos\beta} = \tan\gamma$ and $-\pi/2<\alpha, \beta,\gamma<\pi/2$", and then do something piecewise for the other half of the circle. –  Michael Hardy Dec 10 '12 at 14:31
Empirical evidence confirms my comment above. –  Michael Hardy Dec 10 '12 at 18:19
I should have interchanged the "if" and the "then"? Then the proposition would be true without the extra condition that the angles are between $\pm\pi/2$. –  Michael Hardy Jan 21 '13 at 1:34