By sketching suitable graphs, show that the equation
$$\sec x = 3 - x ^ { 2 }$$
has exactly one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
Show that, if a sequence of values given by the iterative formula
$$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x _ { n } ^ { 2 } } \right)$$
converges, then it converges to a root of the equation given in part (i).
Use this iterative formula, with initial value \(x _ { 1 } = 1\), to determine the root in the interval \(0 < x < \frac { 1 } { 2 } \pi\) correct to 2 decimal places, showing the result of each iteration.