Your algebra is correct, you could also arrive there by some analysis.
csc(x) as x -> pi becomes infinitely large.
You cannot directly compute sqrt(x), as x -> pi, since pi is irrational and has no square root.
BUT, since csc(x), x -> pi, becomes huge the closer and closer you get to pi, and since sqrt(x), x->pi, is roughly constant. You have a denominator that is increasingly increasing, and a “constant” numerator. Thus it must tend to 0.
Hope that helps with the reasoning.