Well, csc x = 1/sinx, so (csc x)' = -cos x/sin^2(x). Do you really think this is nastier than -csc x*cot x??? Of course, you save yourself a quotient...
Anyway, the derivation looks fine to me.
The derivative of is quoted in all the reference works and textbooks that I've ever come across as . IMO it's perfectly appropriate, and I actually find it easier to learn like that. As you say, it saves a quotient.
To take it to its logical conclusion, there's no point in learning it anyway, because you can always use the quotient rule on .