Without using a calculator, show that \(\sinh \left( \cosh ^ { - 1 } 2 \right) = \sqrt { 3 }\).
It is given that, for non-negative integers \(n\),
$$I _ { n } = \int _ { 0 } ^ { \beta } \cosh ^ { n } x \mathrm {~d} x , \quad \text { where } \beta = \cosh ^ { - 1 } 2$$
Show that \(n I _ { n } = 2 ^ { n - 1 } \sqrt { 3 } + ( n - 1 ) I _ { n - 2 }\), for \(n \geqslant 2\).
Evaluate \(I _ { 5 }\), giving your answer in the form \(k \sqrt { 3 }\).