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##