(a)[2]
Sketch an appropriate pair of graphs to show that the equation $\cot \frac{1}{2}x = 1 + e^{-x}$ has exactly one root in the interval $0 < x \leq \pi$.
(b)[2]
Verify through calculation that this root is between $1$ and $1.5$.
(c)[3]
Use the iterative formula $x_{n+1} = 2\tan^{-1}\left(\frac{1}{1 + e^{-x_n}}\right)$ to find the root correct to $2$ decimal places. Give each iterative value to $4$ decimal places.