Perform the subtraction and use the fundamental identities to simplify

Question

Perform the subtraction and use the fundamental identities to simplify. There is more than one correct form of the answer.

tan x − sec2 x/tan x

 

Answer 1

tanx - (secx)^2/tanx = tanx*tanx/tanx - (secx)^2/tanx = (tanx)^2/tanx - (secx)^2/tanx

= [(tanx)^2 - (secx)^2]/tanx = [(tanx)^2- (1 + (tanx)^2)]/tanx

[(tanx)^2 - 1 - (tanx)^2]/tanx = -1/tanx = -cotx.