The foregoing discussion is related to the Riactti transformation for differential equations, wherein a linear system is converted into a nonlinear system (actually, a second order system) by introducing a quotient of the linear solutions. The quotient would also be the basis for a continued fraction exposition of the properties of the solutions of the recursion equation, but nowadays matrix theory is more familiar than continued fraction theory which is why it is used instead.
Nevertheless, the graph of the quotient which figures in the development just outlined is very reminiscent of the graph of the tangent of a multiple angle, leading to the question: why not use the angle itself? This would be analogous to using the Prüfer transformation from differential equation theory, and is not so far from the appearance of hyperbolic cosines in the dispersion relation.