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In this recent question, the equation $$\tan\left(\frac{1}{A}\right) = \tan\left(\frac{1}{B}\right) + \tan\left(\frac{1}{C}\right)$$ is said to imply $$A + B + C = ABC$$ without any stated constraints. Where does this come from, and is it even correct?

I must be misunderstanding the context here, because even with integers like $B = 4$ and $C = 5$, the equations disagree.

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    Even the solution in integers ${A=3,B=2,C=1}$ of $A+B+C=ABC$ fails to satisfy the tangent equation. – GEdgar Aug 01 '14 at 15:45

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