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Well, you can start by testing a couple of values, to disprove it immediately try A=30 and B=59 Tan30 + tan59 = tan89 ?
I for one, do not recall the tangent identity to be quite as simple as tan(A) + tan(B) = tan(A + B)...

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Other answers:

It's not @terenzreignz, but the question states for A+B< 90 which makes it not able to be an identity. But it's easy to disprove.
Why not tan(45) + tan(30) = tan(75) :)
tan 45 = 1. tan 22.5 <> .5
^that also disproves it, @highschoolmom2010
basically tana+tanb= (sinacosb+sinbcosa)/cosacosb= sin(a+b)/cosacosb
At least we have exact values for these stuff :3 \[\Large \tan(45^o) = 1\\\Large \tan(30^o)=\frac1{\sqrt{3}}\]
My reasoning for picking a A+B close to 90, was due to tan90 being undefined, so it's highly unlikely two smaller angles will give anything close to the tangent of an angle near 90
=(sinacosb+cosasinb)/cosacosb=/= sinasinb/cosacosb
Noted :)
Now i need to somehow prove sinacosb+sinbcosa=/=sinasinb...
@katherine.ok based on her other questions, I don't think she's at the level of trigonometry where they're using proofs. I think it's more an analytical question.
tan 60 + tan (-30) <> tan 30
I think as simple as this sound...
by using the tangent graph, adding a random point where x is (0,90),you will not get tan a+ tan b because it is not linear graph.
^ good point.
im lost on all of this tbh....

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