Prove cotx-1/cotx+1 = sec2x - tan2x

Question

I prove till cotx-1/cotx+1 =1/1+tanx - tanx/1+tanx

Answer 1

For this to work you must mean
(cotx-1)/(cotx+1) = sec(2x) - tan(2x)

LS = (cosx/sinx - 1)/(cosx/sinx +1)
multiply top and bottom by sinx
= (cosx - sinx)/(cosx + sinx)

RS = 1/cos(2x) - sin(2x)/cos(2x)
= (1 - 2sinxcosx)/(cos^2 x - sin^2 x)
= (sin^2 x + cos^2 x - 2sinxcosx)/((cosx+sinx)(cosx-sinx))
= (sinx - cosx)^2/( (cosx+sinx)(cosx-sinx))
= (sinx - cosx)/(cosx + sinx)
= LS