Trigonometry question?

Question

Prove sec x/sec x + tan x + tan x / sec x- tan x = 1+ 2 tan^2x

Answer 1

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

Start from left hand side :

Rewrite the expression with common denominator.

Substitute Trigonometric identity : sec² x - tan² x = 1

( sec² x - tan² x ) + 2tan²x

Substitute Trigonometric identity : sec² x - tan² x = 1

1 + 2tan²x

Hence Trigonometric identity sec x/(sec x + tan x) + tan x / ( sec x- tan x )= 1+ 2 tan²x is proved.