(tan x )/(1 + cos x) + (sin x)/(1 - cos x) = cot x + sec x csc x

Starting with the left side:

Get the LCD and add the fractions to get:

[tan x (1 - cos x) + sin x (1 + cos x)] / (1 + cos x)(1 - cos x)

Distributive property:

(tan x - tan x cos x + sin x + sin x cos x) /( 1 - cos^{2}x)

Simplify.

tan x - sin x + sin x + sin x cos x / sin^{2}x

(tan x + sin x cos x) / (sin^{2}x)

Now working with the right side.

cot x + sec x csc x

cos x/sin x + 1/(cos x sin x)

Find the LCD and add to get:

(cos^{2}x + 1) / (sin x cos x)

(1 - sin^{2}x + 1)/ (sin x cos x)

(2 - sin^{2}x) / (sin x cos x)

From here, I cannot find how to relate the two sides of the equation. Please help!