By sketching a suitable pair of graphs, show that the equation \(\cot \frac { 1 } { 2 } x = 1 + \mathrm { e } ^ { - x }\) has exactly one root in the interval \(0 < x \leqslant \pi\).
Verify by calculation that this root lies between 1 and 1.5.
Use the iterative formula \(x _ { n + 1 } = 2 \tan ^ { - 1 } \left( \frac { 1 } { 1 + \mathrm { e } ^ { - x _ { n } } } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.