The Quotient Rule for derivatives states:
If f(x) = g(x)/(h(x))
then
f'(x) = (g'(x)h(x) - g(x)h'(x))/(h(x))^2
SEE: https://en.wikipedia.org/wiki/Quotient_rule
So, the first step is to identify g(x) and h(x) in your original function:
f(x) = sin(x)/(1- cos(x))
so
g(x) = sin(x)
h(x) = 1- cos(x)
We then differentiate the two components of f(x) with respect to x:
g'(x) = cos(x)
h'(x) = -(-sin(x)) = sin(x)
Let's combine all of the pieces, based on the Quotient Rule, and then simplify:
f'(x) = (g'(x)h(x) - g(x)h'(x))/(h(x))^2
f'(x) = (cos(x)(1- cos(x)) - sin(x) sin(x))/(1- cos(x))^2
f'(x) = (cos(x) - cos^2(x) - sin^2(x))/(1- cos(x))^2
f'(x) = (cos(x) - (cos^2(x) + sin^2(x)))/(1- cos(x))^2
f'(x) = (cos(x) - 1)/(1- cos(x))^2
f'(x) = (cos(x)- 1)/((-1)(1- cos(x))(-1)(1- cos(x)))
f'(x) = cancel(cos(x) - 1)/(cancel((cos(x) - 1))*(cos(x) - 1)
f'(x) = 1/(cos(x) - 1)