Why do I keep getting stuck when taking the derivative of the inverse cosecant function?

Hello all,

So I’m attempting to directly prove that the derivative of the inverse cosecant: (d/dx) csc^-1(x) = -1 / [abs(x) sqrt(x^2 – 1)]

To get started, I swap x and y in the cosecant function expression y = csc^-1(x), and get:

x = csc(y)

Then I go ahead and take the derivative of that expression, and I get:

1 = -csc(y)cot(y) y’

So,

y’ = – 1 / [csc(y)cot(y)]

How do I change csc(y)cot(y) to the required abs(x) sqrt(x^2 -1)? Is this some bizarre trigonometric/Pythagorean identity of which I am not aware?