By sketching a suitable pair of graphs, show that the equation
$$2 \cot x = 1 + \mathrm { e } ^ { x }$$
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 1.0 .
Show that this root also satisfies the equation
$$x = \tan ^ { - 1 } \left( \frac { 2 } { 1 + \mathrm { e } ^ { x } } \right)$$
Use the iterative formula
$$x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 2 } { 1 + \mathrm { e } ^ { x _ { n } } } \right) ,$$
with initial value \(x _ { 1 } = 0.7\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.