(a)[4]
Express $\frac{dy}{dx}$ in terms of $\tan x$, and check that when $x = \tfrac{1}{4}\pi$ the result is $\frac{dy}{dx} = 1$.
(b)[4]
At the other point on the curve where $x=a$, the value of $\frac{dy}{dx}$ is also $1$, as the diagram shows. Show that $t^3 + t^2 + 3t - 1 = 0$, where $t = \tan a$.
(c)[3]
Use the iterative formula $a_{n+1} = \tan^{-1}\left( \tfrac{1}{3}(1 - \tan^2 a_n - \tan^3 a_n) \right)$ to find $a$ correct to 2 decimal places, with each iterate shown to 4 decimal places.