(a)[2]
Using sketches of a suitable pair of graphs, show that the equation $\sec 2x = -e^{-x}$ has just one root in the interval $0 < x < \frac{1}{2}\pi$.
(b)[2]
Show by calculation that this root is between $0.9$ and $1$.
(c)[1]
Show that if a sequence of values produced by the iterative formula $x_{n+1} = \frac{1}{2}\cos^{-1}\!\left(-e^{-x_n}\right)$ converges, then its limit is the root of the equation in part (a).
(d)[3]
Use the iterative formula from part (c) to find $x$ correct to 3 decimal places. Record the result of each iteration to 5 decimal places.