Proving trigonometric identities Prove: sin2A = 2tanA/1 + tan^2A

We need to prove that:

sin2A
= 2tanA/ (1+tan^2 A)

We will start from the right side and
prove the left side.

We know that tanA =
sinA/cosA

==> 2tanA / (1+tan^2 A) = 2(sinA/cosA) /
[1+ (sinA/cosA)^2]

                                = 2sinA/
cosA[ 1+ (sin^2A/cos^2
A)]

                                = 2sinA/ cosA*[(cos^2 A
+ sin^2 A)/cos^2A]

But we know that sin^2 A + cos^2 A =
1

==> 2tanA/ (1+tan^2 A) = 2sinA/ 
(1/cosA)

                                      =
2sinA*cosA

But we know that sin2A =
2sinA*cosA

==> 2tanA / (1+tan^2 A) =
sin2A