Trig Identities

Question

Prove

Sin^(2x)tanx=sin(2x)-sin(2x)cos(2x)

Answer 1

right hand side:

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

= sin(2x)[sin^2(x) + cos^2(x) - (cos^2(x) - sin^2(x))] =

= sin(2x)[2sin^2(x)] =

= sin(2x)[2 sin(x)cos(x) * sin(x)/cos(x)] = (multiply and divide by cosx)

= sin(2x) * sin(2x) * tan(x)

QED

The following identities were used: sin^2(x) + cos^2(x) = 1, sin(2x) = 2sinxcosx, cos(2x) = cos^2(x) - sin^2(x)