Trigonometry question

Question

Prove the identity. Cot(2x)=1-tan^2(x)/2tan(x)

Answer 1

sin2x = 2sinxcosx

cos2x = cos^2x - sin^2x

tan2x = sin2x/cos2x = 2sincosx/(cos^2x - sin^2x)

cot2x = 1/tan2x = (cos^2x-sin^2x)2sinxcosx = cos^2x/2sinxcosx - sin^2x/2sinxcosx = cosx/2sinx - sinx/2cosx =

= cotx/2 - tanx/2 = (1/2tanx -tanx/2 = 1/2tanx - tan^2x/2tanx= (1-tan^2x)/2tanx

cot2x = (1-tan^2(x))/2tan(x)