Question

Prove that: `1/(sec x - tan x) - 1/(cos x) = 1/(cos x) - 1/(sec x + tan x)`

Answer 1

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

LHS = `1/(sec x - tan x) + 1/(sec x + tan x)`

= `((sec x + tan x) + (sec x - tan x))/((sec x - tan x)(sec x + tan x))`

= `(2 sec x)/(sec^2x - tan^2x)`

Since sec² x − tan² x = 1

`2 sec x = 2/(cos x)`

RHS = `1/(cos x) + 1/(cos x) = 2/(cos x)`

LHS = RHS.

Hence proved.