How do you prove (2tanx)/ (1 + tan² x) = sin 2x?

We have to prove that (2tanx)/ (1+tan^2 x) = sin 2x

Solution

Let us start with RHS

(2tanX)/(1+tan^2X)

= 2/[1/tan(x)=tan(x)]

= 2/[(sin²(x}+cos²(x))/(sin(x)cos(x))]

= 2/[1/(sin(x)cos(x))]

= 2(sin(x)cos(x)

=sin(2x)

= RHS

Hence Proved