Question

How do you prove the identity `(tan x + sec x - 1) / (tan x - sec x +1) = tan x + sec x`?

Answer 1

`sin^2 A + cos^2 A = 1`
Divide by `cos^2 A`:
`tan^2 A + 1 = sec^2 A`

`(tan x + sec x - 1)/(tan x - sec x + 1)`
`= (tan x + sec x - 1)^2/((tan x + sec x - 1)(tan x - sec x + 1)`
`= (tan^2 x + 2tan x(sec x - 1) + (sec x - 1)^2)/(tan^2 x - (sec x - 1)^2`
`= (tan^2 x + 2tan x(sec x - 1) + (sec^2 x - 2sec x + 1))/(tan^2 x - (sec^2 x -2sec x + 1))`
`= (2tan x(sec x - 1) + 2sec x(sec x - 1))/(2(sec x - 1))`
`= tan x + sec x`