Question

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

Answer 1

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`