sin x / (1 - cos x) + tan x / ( 1 + cos x) = multiply both parts of the first fraction by 1 + cos x and both parts of the second fraction by 1 - cos x; common denominator is 1 - cos^2 x = sin^2 x

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

(sin x + sin x cos x + tan x - sin x) / (sin^2 x) =

(sin x cos x + sin x / cos x) / sin^2 x = (cos x + 1 / cos x) / sin x = (cos^2 x + 1) / (sin x cosx)