The hyperbolic functions are called hyperbolic because $$ \cosh^2(t)-\sinh^2(t)=1 $$gives us a hyperbola under the transformation $$ x= \cosh(t)\\ y=\sinh(t) $$
Find two hyperbolic functions which give us a circle and find two trig functions which give us a hyperbola.
Are true hyperbolic functions only $\cosh(x)$ and $\sinh(x)$?
Just take the standard identities and divide by either $\cos^2 \theta$ or $\cosh^2 t \; \; ,$ giving
$$ \sec^2 \theta - \tan^2 \theta = 1 \; , $$
$$ \tanh^2 t + \operatorname{sech}^2 t = 1. $$
The second one is important, with $B > 0$ $$ \left( A + B \tanh t \; , \; \; B \operatorname{sech} t \right) $$ gives a unit-speed geodesic in the upper half plane model of the hyperbolic plane. The other unit speed geodesics are just $(A, e^t).$ In both cases we can move the starting point, when convenient, by substituting $t- t_0$ for $t.$