Prove that the following equation is an identity:

cot(x)/[1+tan(-x)] + tan(x)/[1+cot(-x)] = cot(x) + tan(x) + 1

cot(x)/[1+tan(-x)] + tan(x)/[1+cot(-x)]

= cot(x)/[1 - tan(x)] + tan(x)/[1 - cot(x)]

multiply top and bottom of second fraction with tan x

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

= cot(x)/[1 - tan(x)] - tan^2(x)/[1 - tan x)]

take common LCM

= ( cot x - tan^2(x)) /(1 - tan x)

=( (1 / tan x) - tan^2(x)) /(1 - tan x)

= ( 1 - tan^3(x)) / tan x(1 - tan x)

use the formula for difference of cubes

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

= (1 + tan x + tan^2(x)) /tan x

= (1 / tan x) + (tan x / tan x) + ( tan^2(x) / tan x)

= cot x + 1 + tan x

cot(x)/[1+tan(-x)] + tan(x)/[1+cot(-x)]
= cot(x)/[1-tan(x)] + tan(x)/[1-cot(x)]
= cot(x)/[1-tan(x)] - tan^2(x)/[1-tan(x)]
= cot(x)(1-tan^3(x))/[1-tan(x)]
= cot(x)(1 + tan(x) + tan^2(x))
= cot(x) + tan(x) + 1