The curve \(C\) has equation \(y = 2 \sec x\), for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). Show that the arc length \(s\) of \(C\) is given by $$S = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( 2 \sec ^ { 2 } x - 1 \right) d x$$ Find the exact value of \(s\). The surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(S\). Show that

1. \(S = 4 \pi \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( 2 \sec ^ { 3 } x - \sec x \right) \mathrm { d } x\),
2. \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sec x \tan x ) = 2 \sec ^ { 3 } x - \sec x\). Hence find the exact value of \(S\).