Equation of Catenary/Formulation 2/Proof

Proof
Take the definition of the catenary according to Formulation 1:
 * $y = \dfrac {e^{ax} + e^{-ax}} {2 a}$

Put this in a form which uses the hyperbolic cosine:
 * $y = \dfrac {\cosh a x} a$

Replace $a$ with $\dfrac 1 a$:
 * $y = a \cosh \dfrac x a$

Hence the result.