Question

prove that

_______1______ - ___1___ =__1___
sec x-tan x cos x cos x
- _______1______
sec x + tan x

Answer 1

I will rewrite the question as:

1/(sec(x)-tan(x)) - 1/cos(x) = 1/(cos(x) - 1/(sec(x)+tan(x))

transpose terms to get:
1/(sec(x)-tan(x)) + 1/(sec(x)+tan(x)) = 2/cos(x)

Add the two fractions on the left and simplify using sec²(x)=1+tan²(x) on the left-hand side to get 2sec(x)=2/cos(x).