What is cosh⁡, sinh⁡, tanh⁡?

Example 4.20 (The addition formula for $\tanh$)

Dividing the two addition formulas of Proposition 4.18 (3) by $\cosh x\cosh y$:

the hyperbolic sibling of the tangent addition formula — with a $+$ where trigonometry has a $-$. A dividend: since $\abs{\tanh} < 1$, the right-hand side is a “velocity addition” rule that never leaves $\intoo{-1}1$: if $u, v \in \intoo{-1}1$ then $\frac{u + v}{1 + uv} \in \intoo{-1}1$ as well (write $u = \tanh a$, $v = \tanh b$, possible by bijectivity, and read the formula backwards). Checking that algebraically, without hyperbolic functions, is a slightly painful exercise; parametrizing by $\tanh$ makes it one line — the same strategy that circular functions provide for the unit circle.

Example 4.22 (Closed forms at work)

The solution of $\cosh t = 2$ with $t \geq 0$ is, by the closed form, $t = \operatorname{arcosh} 2 = \ln(2 + \sqrt3) \approx 1.317$; the other solution is $-t$, by evenness — and indeed $\ln(2 - \sqrt3) = \ln\frac1{2 + \sqrt3} = -\ln(2 + \sqrt3)$: the two roots $u = \eu^t$ of the quadratic $u^2 - 4u + 1 = 0$ are reciprocals, as their product $1$ (Vieta) demands. This tiny computation displays the general pattern: hyperbolic equations convert to quadratics in $\eu^t$, and the symmetry $t \mapsto -t$ appears as the symmetry $u \mapsto 1/u$ of the quadratic — worth remembering when solving Exercise 4.7.