Starting from the definitions of cosh and sinh in terms of exponentials, prove that
$$2 \sinh ^ { 2 } A = \cosh 2 A - 1$$
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A curve has equation \(\mathrm { y } = \mathrm { x } ^ { 2 }\), for \(0 \leqslant x \leqslant \frac { 2 } { 3 }\). The area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(S\).
Use the substitution \(\mathrm { X } = \frac { 1 } { 2 } \operatorname { sinhu }\) to show that \(S = \frac { 1 } { 32 } \pi \left( \frac { 820 } { 81 } - \ln 3 \right)\).