The given identity is false.

##(sec(x) – tan(x))(sec(x) + tan(x)) = 1##

We will be using the following:

##sec(x) = 1/cos(x)## (by definition)

##tan(x) = sin(x)/cos(x)## (by definition)

##(a-b)(a+b) = a^2 – b^2## (difference of squares formula)

##sin^2(x) + cos^2(x) = 1## (identity)

##(sec(x) – tan(x))(sec(x) + tan(x)) = sec^2(x) – tan^2(x)##

(by the difference of squares formula)

##= (1/cos(x))^2 – (sin(x)/cos(x))^2##

(by definition of secant and tangent)

##= 1/cos^2(x) – sin^2(x)/cos^2(x)##

##=(1 – sin^2(x))/cos^2(x)##

##= cos^2(x)/cos^2(x)##

(by the above identity)

##=1##