(a)[2]
By plotting a suitable pair of graphs, demonstrate that the equation $\cot 2x = \sec x$ has a single root in the interval $0 < x < \frac{1}{2}\pi$.
(b)[1]
Show that if a sequence of real numbers defined by the iterative rule $x_{n+1} = \frac{1}{2}\tan^{-1}(\cos x_n)$ converges, then its limit is the root in part (a).