By sketching a suitable pair of graphs, show that the equation
$$\cot x = 4 x - 2$$
where \(x\) is in radians, has only one root for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
Verify by calculation that this root lies between \(x = 0.7\) and \(x = 0.9\).
Show that this root also satisfies the equation
$$x = \frac { 1 + 2 \tan x } { 4 \tan x }$$
Use the iterative formula \(x _ { n + 1 } = \frac { 1 + 2 \tan x _ { n } } { 4 \tan x _ { n } }\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.