Question 8 - A-Level Maths

Define tanh $y$ in terms of $\mathrm { e } ^ { y }$ and $\mathrm { e } ^ { - y }$.
Given that $y = \tanh ^ { - 1 } x$, where $- 1 < x < 1$, prove that $y = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)$.
Find the exact solution of the equation $3 \cosh x = 4 \sinh x$, giving the answer in terms of a logarithm.
Solve the equation $$\tanh ^ { - 1 } x + \ln ( 1 - x ) = \ln \left( \frac { 4 } { 5 } \right)$$