Question 7 - A-Level Maths

Use the definitions $$\sinh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } \right) \quad \text { and } \quad \cosh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } \right)$$ to show that:
1. $2 \sinh \theta \cosh \theta = \sinh 2 \theta$;
2. $\cosh ^ { 2 } \theta + \sinh ^ { 2 } \theta = \cosh 2 \theta$.
A curve is given parametrically by $$x = \cosh ^ { 3 } \theta , \quad y = \sinh ^ { 3 } \theta$$
1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} \theta } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} \theta } \right) ^ { 2 } = \frac { 9 } { 4 } \sinh ^ { 2 } 2 \theta \cosh 2 \theta$$
2. Show that the length of the arc of the curve from the point where $\theta = 0$ to the point where $\theta = 1$ is $$\frac { 1 } { 2 } \left[ ( \cosh 2 ) ^ { \frac { 3 } { 2 } } - 1 \right]$$