(a)[2]
Using sketches of an appropriate pair of graphs, show that the equation $\sec 2x = -2x - \frac{1}{2}$ has precisely one root in the interval $0 \le x \le \frac{\pi}{2}$.
(b)[2]
Show by calculation that this root is located between $0.8$ and $1.2$.
(c)[2]
Show that, if a sequence of real values defined by the iterative formula $x_{n+1} = \frac{1}{2}\cos^{-1}\!\left(\frac{-2}{4x_n + 1}\right)$ converges, then its limit is the root of the equation in part (a).
(d)[3]
Use this iterative formula to find this root correct to $3$ decimal places. Give each iteration result to $5$ decimal places.