By sketching a suitable pair of graphs, show that the equation \(\cot 2 x = \sec x\) has exactly one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
Show that if a sequence of real values given by the iterative formula
$$x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \cos x _ { n } \right)$$
converges, then it converges to the root in part (a).