(a)[2]
By sketching an appropriate pair of graphs, show that the equation $\cosec x = 1 + e^{-\frac{1}{2}x}$ has exactly two roots in the interval $0 < x < \pi$.
(b)[3]
The sequence produced by $x_{n+1} = \pi - \sin^{-1}\left(\frac{1}{e^{-\frac{1}{2}x_n} + 1}\right)$, with initial value $x_1 = 2$, approaches one of these roots. Use the formula to find this root correct to 2 decimal places. State each iterate to 4 decimal places.