And: Sin(A+B)=SinA*CosB+SinB*CosA.

Sin(A-B)=SinA*CosB-SinB*CosA substitution, finishing can be obtained:

a^2/b^2=tanA/tanB - ( 1)

By sine theorem: a/b = sina/sinb-(2)

The simultaneous expression (1) (2) can be obtained: sin (2a) = sin (2b).

So: 2A=2B or 2a = 180-2b.

So: A=B or a+b = 90.

Therefore: ABC is an isosceles triangle or a right triangle.