(i)[2]
By sketching an appropriate pair of graphs, show that the equation $\cot x = 4x - 2$, where $x$ is in radians, has a single root for $0 \leq x \leq \frac{\pi}{2}$.
(ii)[2]
Verify by calculation that this root is between $x = 0.7$ and $x = 0.9$.
(iii)[1]
Show that this root also satisfies the equation $x = \frac{1 + 2\tan x}{4\tan x}$.
(iv)[3]
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 value of each iteration to $4$ decimal places.