(i) By sketching a suitable pair of graphs, show that the equation \(\cot x = 1 + x^2\), where \(x\) is in radians, has only one root in the interval \(0 < x < \frac{1}{2}\pi\).

(ii) Verify by calculation that this root lies between 0.5 and 0.8.

(iii) Use the iterative formula \(x_{n+1} = \arctan\left( \frac{1}{1 + x_n^2} \right)\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.