Prove cosx/(1-tanx) - cosx = sin x - sinx/(1-cotx)

For the beginning, we'll re-arrange the terms of the
given expression:

cos x/(1-tan x) - cos x = sin x - sin
x/(1-cot x)

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

We'll re-write the terms from the left side of the
given expression:

cos x/(1-tan x) = cos x/(1- sin x/cos
x)

cos x/(1- sin x/cos x) = cos x/[(cos x-sin x)/cos
x]

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

We'll re-write the terms from the right side of the
given expression:

sin x/(1-cot x) = sin x/(1- cos x/sin
x)

sin x/(1- cos x/sin x) = (sin x)^2/(sin x-cos x)
(2)

We'll add (1) and
(2):

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

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

We'll re-write the difference of
squares:

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

The left side of the expression will
become:

(cos x - sin x)(cos x + sin x)/(cos x-sin
x)

We'll reduce the like
terms:

(cos x - sin x)(cos x + sin x)/(cos
x-sin x) = cos x + sin x