Analogies between $(\tan, \sec)$ and $(\sinh, \cosh)$ The pair of functions $(\tan, \sec)$ shares some interesting properties with the pair $(\sinh, \cosh)$.

First of all, they satisfy the same quadratic equation, namely
$$\sec^2 x - \tan^2 x = 1 \qquad \cosh^2 x - \sinh^2 x = 1$$
for any $x$ in the respective domains.

Moreover, $\tan$ and $\sinh$ are both odd functions, while $\sec$ and $\cosh$ are both even functions.

Now, suppose that we define a binary operation $\oplus$ on some subset of real numbers such that
$$\tan (x \oplus y) = \tan x \sec y + \sec x \tan y$$
whenever $x \oplus y$ is defined. Then one can prove that
$$\sec(x \oplus y) = \sec x \sec y + \tan x \tan y$$
and these two formulas look exactly like the addition formulas for the hyperbolic functions. (For the subtraction formulas it is enough to let $x \ominus y = x \oplus (-y)$ whenever it is defined.)

There's more: one can also prove that an analogue to De Moivre's formula holds, i.e.,
$$(\sec x + \tan x)^n = \sec (\mathring n x) + \tan (\mathring n x)$$
where $\mathring n x$ denotes $x \oplus x \oplus \dotsb \oplus x$ with $n$ addends. Finally, if we define an analogue of the derivative with this new operation by letting
$$\mathring D f(x) = \lim_{h \to 0} \frac {f(x \oplus h) - f(x)} h$$
then we obtain
$$\mathring D \tan x = \sec x \qquad \mathring D \sec x = \tan x$$
similarly to what happens with the hyperbolic functions.

My questions are:


  Is there a way to make this correspondence precise so that one can give a simple unique explanation for all of these analogies (and possibly others that might hold)?
  
  How can we interpret the operation $\oplus$?
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
$$ t\in(-\pi/2,\pi/2) \mapsto(\tan t, \sec t) \qquad \qquad t\in\mathbb R \mapsto (\sinh t, \cosh t) $$
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
$$ \sinh \circ f = \tan \qquad\qquad \cosh \circ f = \sec $$
It's just a particular transformation of the horizontal axis that make the functions into each other.

This means that we have $x\oplus y = f^{-1}(f(x)+f(y))$; 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.
$$\cos(z)=\frac{e^{iz}+e^{-iz}}{2}=$$
$$\cosh(iz)$$

by the same,

$$\sin(z)=\frac{e^{iz}-e^{-iz}}{2i}=$$
$$-i\sinh(iz)$$

To find hyperbolic formulas from trogonometric ones, you just replace

$$\cos \text{ by } \cosh$$
and
$$\sin \text{ by }  -i\sinh$$

For example
$$1-\cos(x)=2\sin^2(\frac{x}{2})$$ becomes

$$1-\cosh(x)=-2\sinh^2(\frac{x}{2})$$