We'lll manage only the left side and we'll re-write the terms of the sum:
csc x = 1/sin x and cot x = cos x/sin x
csc x + cot x = 1/sin x + cos x/sin x
Since the fractions have the same denominator, we can write:
csc x + cot x = (1+cos x)/sin x
The expression will become:
(1+cos x)/sin x = sin x/(1 - cos x)
We'll cross multiply:
(sin x)^2 = (1+cos x)(1-cos x)
The product from the right side returns the difference of two squares:
(sin x)^2 = 1 - (cos x)^2
We'll add (cos x)^2 both sides:
(sin x)^2 + (cos x)^2 = 1
But Pythagorean identity states that (sin x)^2 + (cos x)^2 = 1, therefore we've came up with a true statement.
Therefore, the given identity csc x + cot x = sin x/(1 - cos x) is verified.