Your equation is formatted ambiguously. There are 4 things you could mean:
(1+tanΘ)/(1+tanΘ)=tanΘ. In this case, the left-hand-side is just something divided by itself, which is just 1. So we have 1=tanΘ. This is not true in general, and so is not an identity.
1+(tanΘ)/(1+tanΘ)=tanΘ. In this case, just check Θ=0. We get 1+(0)/(1+0)=0, or 1+0=0, or 1=0. So this is not an identity either, since it's not true in general.
(1+tanΘ)/(1) +tanΘ=tanΘ. Subtract tanΘ from both sides and get (1+tanΘ)/1=0, or 1+tanΘ=0, or tanΘ= -1. Again, not true in general.
1+(tanΘ)/(1) +tanΘ=tanΘ. Subtract tanΘ again, we get 1+(tanΘ)/(1)=0, or 1+tanΘ=0, or tanΘ= -1. Again, not an identity.
So no matter how you meant it, what you have is not an identity.
