Using the definitions for \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that \(\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1\).
Hence solve the equation \(\sinh ^ { 2 } x = 5 \cosh x - 7\), giving your answers in logarithmic form.