Prove that sin2x=2tanx/1+tan2x

Question

We know that

Tan (x+y)=tanx+tany/1+tanx tany

Put y =x

Tan(X+x)= tanx+tanx /1-tanx tany

Tan2x= 2tanx/1-tan2x

Answer 1

To prove sin2x = \(\frac{2tanx}{1+tan^2x}\)

Let us start from RHS and prove it equal to LHS.

RHS = \(\frac{2tanx}{1+tan^2x}\) 

⇒\(\frac{2tanx}{sec^2x}\), as 1 + tan²x  = sec² x,identity

⇒ \(\frac{2\Big(\frac{sinx}{cosx}\Big)}{\frac{1}{cos^2x}}\)

⇒2sinxcosx = sin2x as per sin2x=2sinxcosx identity