Use the definitions \(\cosh x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\) and \(\sinh x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } \right)\) to show that
$$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$
Express
$$5 \cosh ^ { 2 } x + 3 \sinh ^ { 2 } x$$
in terms of \(\cosh x\).
Sketch the curve \(y = \cosh x\).
Hence solve the equation
$$5 \cosh ^ { 2 } x + 3 \sinh ^ { 2 } x = 9.5$$
giving your answers in logarithmic form.