(i)[2]
By drawing a suitable pair of graphs, demonstrate that the equation $\sec x = 3 - \frac{1}{2}x^2$, where $x$ is measured in radians, has a root in the interval $0 < x < \frac{1}{2}\pi$.
(ii)[2]
Use calculation to confirm that this root lies between $1$ and $1.4$.
(iii)[1]
Show that this root also meets the equation $x = \cos^{-1}\left(\frac{2}{6 - x^2}\right)$.
(iv)[3]
Apply an iterative formula based on the equation in part (iii) to find the root correct to $2$ decimal places. Show the outcome of each iteration to $4$ decimal places.