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 \(0 < x < \frac { 1 } { 2 } \pi\).
Verify by calculation that this root lies between 0.5 and 0.8.
Use the iterative formula
$$x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 1 + x _ { n } ^ { 2 } } \right)$$
to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.