Question 5 - A-Level Maths

Using the definitions of $\sinh x$ and $\cosh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$, show that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1$$ Deduce that $1 - \tanh ^ { 2 } x \equiv \operatorname { sech } ^ { 2 } x$.
Solve the equation $2 \tanh ^ { 2 } x - \operatorname { sech } x = 1$, giving your answer(s) in logarithmic form.
Express $\frac { 4 } { ( 1 - x ) ( 1 + x ) \left( 1 + x ^ { 2 } \right) }$ in partial fractions.
Show that $\int _ { 0 } ^ { \frac { 1 } { \sqrt { 3 } } } \frac { 4 } { 1 - x ^ { 4 } } \mathrm {~d} x = \ln \left( \frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 } \right) + \frac { 1 } { 3 } \pi$.