Prove the identity: (1 - sec(x))/(1 + sec(x)) = (cos(x) - 1)/(cos(x) + 1)

We have to prove that: (1 - sec(x))/(1 + sec(x)) = (cos(x) -
1)/(cos(x) + 1)

Let's start from the left hand
side

(1 - sec(x))/(1 + sec(x))

use the
fact that sec x = 1/cos x

=> (1 - (1/cos x))/(1 + (1/cos
x))

=> [(cos x - 1)/cos x]/[(cos x + 1)/cos
x]

=> (cos x - 1)/(cos x +
1)

which is the right hand
side

This proves that (1 - sec(x))/(1 + sec(x)) =
(cos(x) - 1)/(cos(x) + 1)