11. (a) Show that $$\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) , \quad \text { where } - 1 < x < 1$$ (b) Given that $$a \cosh x + b \sinh x \equiv \operatorname { rcosh } ( x + \alpha ) , \quad \text { where } a > b > 0$$ show that $$\alpha = \frac { 1 } { 2 } \ln \left( \frac { a + b } { a - b } \right)$$ and find an expression for \(r\) in terms of \(a\) and \(b\).
(c) Hence solve the equation $$5 \cosh x + 4 \sinh x = 10$$ giving your answers correct to three significant figures.