(a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that
\(2 \sinh ^{2} A=\cosh 2 A-1\)
(b) A curve has equation \(y=x^{2}\), 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 \(x=\frac{1}{2} \sinh u\) to show that \(S=\frac{1}{32} \pi\left(\frac{820}{81}-\ln 3\right)\).