In order for ANY function, trigonometric or not, to have an inverse function, it must be one-to-one. That is, it must not happen that two different points in the domain have the same image in the range, because if it does, the inverse is NOT a function.
The problem is that NONE of the trigonometric functions are one -to-one. So, in order to define the inverse trigonometric functions, we must restrict their domain to a set in which the function IS one-to-one.
In this video, I show you how to do that with the cosecant function and then we define the inverse cosecant specify its domain and range, and go over a couple of properties. We also do a couple of exercises