How do you verify the identity (2tanx)/(1+tan^2x) =sin2x?

Use the fact that:
##tanx=sinx/cosx##
and
##sin2x=2sinxcosx##
So:
##2sinx/cosx*1/(1+sin^x/cos^2x)=2sinxcosx##
##2sinx/cosx*cos^2x/(cos^2x+sin^2x)=2sinxcosx##
##2sinx/cancel(cosx)*cos^cancel(2)x/(cos^2x+sin^2x)=2sinxcosx##
But ##sin^2x+cos^2x=1##
So:
##2sinxcosx=2sinxcosx##