Question

Why are these the same, isn’t the hyperbolic functions related to e only ?

Answer 1

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Answer 2

They’re called hyperbolic trig functions for a reason, because they map to the unit hyperbolas.

Answer 3

Functions like these satisfy a sort of "parametric" form. Cosh^2 x - Sinh^2 x =1 can also be written compactly as x^2 - y^2 =1 (by substituting x = Coshx and y = Sinhx)

Now, if you solve algebraicly for x as an example, you end up with x = ±sqrt( 1+ y^2 ).
Congrats, you just went from an explicit equation to an implicit form.

Answer 4

cosh^2 t - sinh^2 t = 1

Answer 5

cosh^(2)(t)-sinh^(2)(t)=1

Set x=sinh(t), y=cosh(t).