Proof Of The Derivative Of Arccosecant(x)

First lets start with a triangle where one of the angles is y. Lets say the hypotenuse side is of length x and that the adjacent side is of length 1:

Using the triangle above, we can say:

We already know that the derivative of x with respect to x is 1:

Since x = cosec(y):

Using the chain rule:

And we already know what cosec(y) and cotan(y) is equal to: