Question 4 - A-Level Maths

Starting from the definitions of $\sinh x$ and $\cosh x$ in terms of exponentials, prove that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$
Solve the equation $4 \cosh ^ { 2 } x + 9 \sinh x = 13$, giving the answers in exact logarithmic form.
Show that there is only one stationary point on the curve $$y = 4 \cosh ^ { 2 } x + 9 \sinh x$$ and find the $y$-coordinate of the stationary point.
Show that $\int _ { 0 } ^ { \ln 2 } \left( 4 \cosh ^ { 2 } x + 9 \sinh x \right) \mathrm { d } x = 2 \ln 2 + \frac { 33 } { 8 }$.