Analogies between $(tan, sec)$ and $(sinh, cosh)$

I think everything follows from your first equation. Since the signs of tan, sec go like the signs of sinh, cosh, this equation tells us that the parametric graphs
$$ tin(-pi/2,pi/2) mapsto(tan t, sec t) qquad qquad tinmathbb R mapsto (sinh t, cosh t) $$
consist of the same points in a different parameterization (in fact it’s the upper branch of a hyperbola).

So if we define $f(x) = sinh^{-1}(tan t)$, then we have
$$ sinh circ f = tan qquadqquad cosh circ f = sec $$
It’s just a particular transformation of the horizontal axis that make the functions into each other.

This means that we have $xoplus y = f^{-1}(f(x)+f(y))$; in other words $oplus$ is just ordinary addition transfered through this bijection.

And this also means that your $mathring D$ is just ordinary differentiation transfered through the bijection too.

The same $f$ will turn also $sin$ and $cos$ into $tanh$ and $operatorname{sech}$, for more correspondences.

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