I will use x instead of theta to make things easier for myself.
You just need to convert them yourself later.
csc(x) - cot(x)
= 1/sin(x) - cos(x)/sin(x)
= (1 - cos(x)) / sin(x)
= (1 - cos(x))(1 + cos(x)) / (sin(x)(1 + cos(x)))
= (1 - cos^2 (x)) / (sin(x)(1 + cos(x)))
= sin^2 (x) / (sin(x)(1 + cos(x)))
= sin(x) / (1 + cos(x))