Show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos 2 x \mathrm {~d} x = \frac { 1 } { 2 }\).
By using an appropriate trigonometrical identity, find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 2 } x \mathrm {~d} x\).
Use the trapezium rule with 2 intervals to estimate the value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec x d x\), giving your answer correct to 2 significant figures.
Determine, by sketching the appropriate part of the graph of \(y = \sec x\), whether the trapezium rule gives an under-estimate or an over-estimate of the true value.