Sin2y/(1 + tan²y) = 2tany

Question

The question is to prove the lefthand side is equal to the right, I've tried different methods and can't seem to figure it out. I'm sure somewhere there's a step I keep messing up and some help would be greatly appreciated. Thank you

Answer 1

This identity has been expressed wrongly because there is no divide sign. If y=45°, and tan(45)=sin(90)=1, so the identity would be:

LHS: sin(90)/(1+tan²(45))=½; RHS: 2tan(45)=2 and the identity is false because ½≠2. 

But sin(90)(1+tan²(45))=2 so LHS=RHS. Now here's the proof:

sin(2y)=2sin(y)cos(y); 1+tan²(y)=sec²(y)=1/cos²(y).

So sin(2y)(1+tan²(y))=2sin(y)cos(y)/cos²(y)=2sin(y)/cos(y)=2tan(y) QED.