The key formulas for analyzing the collision of relativistic particles can all be derived (3) and (4) of §9.4.

Taking the difference of squares leads to the key (7) of §9.4 relating energy, momentum, and (rest) mass, which holds also for massless particles. (3) and (4) of §9.4 leads directly to \begin{equation} \gamma = \cosh\alpha = \frac{E}{mc^2} = \frac{1}{\sqrt{1-\uucsq}} \end{equation} and \begin{equation} \sinh\alpha = \frac{p}{mc} = \uc\,\gamma \end{equation} and dividing these formulas yields \begin{equation} \tanh\alpha = \frac{p\cc}{E} = \uc \end{equation}