Prove, using exponential functions, that
$$\cosh 2 x = 1 + 2 \sinh ^ { 2 } x$$
Differentiate this result to obtain a formula for \(\sinh 2 x\).
Solve the equation
$$2 \cosh 2 x + 3 \sinh x = 3$$
expressing your answers in exact logarithmic form.
Given that \(\cosh t = \frac { 5 } { 4 }\), show by using exponential functions that \(t = \pm \ln 2\).
Find the exact value of the integral
$$\int _ { 4 } ^ { 5 } \frac { 1 } { \sqrt { x ^ { 2 } - 16 } } \mathrm {~d} x$$