Help proving this trigonometric equation as an identity?

Question

[-1/(tan x - sec x)]+ [-1/(tan x + sec x)] = 2 tan x

Answer 1

The identity is [-1/(tan x - sec x)]+ [-1/(tan x + sec x)] = 2 tan x

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

= - 2 tan x / (tan ²x - sec ² x)

= - 2 tan x/-( sec ² x - tan² x )

= -2 tan x/ -1    [Since sec² x -tan ²x = 1]

 = 2 tan x .

Hence the value of the given identity is 2 tan x .