Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that
(a) \(\cosh ( \ln a ) \equiv \frac { a ^ { 2 } + 1 } { 2 a }\), where \(a > 0\),
(b) \(\cosh x \cosh y - \sinh x \sinh y \equiv \cosh ( x - y )\).
Use part (i)(b) to show that \(\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1\).
Given that \(R > 0\) and \(a > 1\), find \(R\) and \(a\) such that
$$13 \cosh x - 5 \sinh x \equiv R \cosh ( x - \ln a )$$
Hence write down the coordinates of the minimum point on the curve with equation \(y = 13 \cosh x - 5 \sinh x\).