Let's say we have the following equation:
$\tan(x+y)=a(\tan(x)+\tan(y))$, where $a$ is a nonzero real number.
What can we say about $x$ and $y$? Is there a general solution?
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If $a=1$ then addition formula for tangent gives $\tan x \tan y=0,$ so one of the tanents must be $0,$ and it seems that's equivalent. But this $a=1$ case is a bit too simple. Still starting with the addition formula one might derive a relation between $x,y$ to characterize your relation.