Show that 2tan(th) / (1+tan^2(th)) = sin(2th), where th = theta

We have 2tan(th) / (1 + tan^2(th)) = sin(2th)

We know that tan(A) = sin(A) / cos(A), and 1 + tan^2(A) = sec^2(A)

Therefore => (2sin(th) / cos(th)) / sec^2(th)

=> 2sin(th)*cos^2(th) / cos(th)

=> 2sin(th) cos(th)

=> sin(2th) by definition of trigonometric identities