(a)[2]
By sketching an appropriate pair of graphs, show that the equation $\cot 2x = 2\sin 2x - 1$ has exactly one root in the interval $0 < x < \frac{1}{2}\pi$.
(b)[2]
Show by calculation that the root is within the interval $0.4 \le x \le 0.6$.
(c)[3]
Use the iterative formula $x_{n+1}=\frac{1}{2}\tan^{-1}\!\left(\frac{1}{2\sin 2x_n-1}\right)$ to find the root correct to 2 decimal places. Record each iterate to 4 decimal places.