# 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.