(i)[2]
By sketching a suitable pair of graphs, show that the equation $\cot x = 1 + x^2$, with $x$ measured in radians, has a single root for $0 < x < \frac{1}{2}\pi$.
(ii)[2]
Verify by calculation that this root is located between $0.5$ and $0.8$.
(iii)[3]
Use the iterative formula $x_{n+1} = \tan^{-1}\left(\frac{1}{1 + x_n^2}\right)$ to find this root correct to $2$ decimal places. Record each iteration to $4$ decimal places.