Here's my work one step at a time.
1) (Cot^2 X)/(1 + Csc X) + (1/Csc X) * Csc X
2) (Cot^2 X)/(1 + Csc X) + 1
3) (Csc^2 X - 1)/(1 + Csc X) + 1
Is this right so far?
Actually after taking a second look, it appears that this would simplify to Csc X.