prove that (tanx-cotx)/(tanx+cotx)=1-2cos^2x

LHS =
(tanx-cotx)/(tanx+cotx))
= (tan x- 1/tan x)/(tan x+ 1/tan x)
= (tan ^2 x-1)/(tan^2 x+ 1)
= (tan ^2 x-1)/ sec^2 x
= cos^2 x( tan ^2 x - 1)
= sin ^2 x - cos ^2 x
= (1- cos^2 x) - cos^2 x = 1- 2 cos^2 x
= RHS proved