Tangent identity given $a + b + c = \pi$ Given that $a + b + c = \pi$, that is, three angles in a triangle - then prove that $$\tan a + \tan b + \tan c = \tan a \tan b \tan c$$
Is my solution below completely rigorous? Can I justify taking the tangent of both sides of my equation (I think not, since tangent isn't an injective function). 

We can write $a + b = \pi -c$ then taking the tangent of both sides, this yields $$\tan (a +b) = \tan(\pi -c) \iff \frac{\tan a + \tan b}{1 - \tan a \tan b} = -\tan c$$
So $$\tan a + \tan b = \tan a \tan b \tan c - \tan c$$
Hence we arrive at $$\bbox[10px, border: blue 1px solid]{\tan a + \tan b + \tan c = \tan a \tan b \tan c} \quad \square$$
as required. 
