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Defining the hyperbolic cosecant function

The hyperbolic cosecant function is an old mathematical function.

This function has a simple definition as the reciprocal to hyperbolic sine functions:

After comparison with the famous Euler formula for the sine function , it is easy to derive the following representation of the hyperbolic cosecant through the circular cosecant:

This formula allows the derivation of all the properties and formulas for the hyperbolic cosecant from the corresponding properties and formulas for the circular cosecant.