Having a problem with an exercise from Gelfand and Saul's *Trigonometry*, in the section dealing with the half-angle formulae. The exercise (7.a. on p.151) asks the reader to show that:

$$\tan(\alpha/2)\tan(\beta/2)\ +\ \tan(\beta/2)\tan(\gamma/2)\ +\ \tan(\gamma/2)\tan(\alpha/2)\ =\ 1$$ 

where $\alpha + \beta + \gamma = \pi$.

I made the cotangent substitution for one of the variables and checked it on Wolfram Alpha, which confirms the identity, but for some reason I still can't see how to actually prove it myself. I'm sure I'm missing something very obvious!