Is the following an identity: csc x + cot x = sin x/1-cos x

We'll re-write the expression using
brackets:

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

Now, we'll multiply both sides by (1-cos
x):

(1 - cos x)(csc x + cot x) = sin
x

We'll remove the brackets using FOIL
method:

cscx + cot x - cos x* csc x - cos x*cot x = sin
x

We'll substitute csc x = 1/sin
x:

1/sin x + cot x - cos x/sin x - cos x*cot x = sin
x

1/sin x + cot x - cot x - cos x*cot x = sin
x

We'll eliminate like terms:

1/sin x -
cos x*cot x = sin x

But cot x = cos x/sin
x

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

Since the fractions have the same denominator, we'll re-write the
left side:

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

But, from Pythagorean identity, we'll
get:

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

The
identity will become:

(sin x)^2/sin x = sin
x

W'll simplify and we'll get:

sin x =
sin x

Since both sides are equal, then the identity
csc x + cot x = sin x/1-cos x is verified, for any real value of
x.