(i)[4]
Given that $2\tan 2x + 5\tan^2 x = 0$, let $\tan x=t$ and form an equation in $t$; hence show that either $t = 0$ or $t = \sqrt[3]{(t + 0.8)}$.
(ii)[2]
It is stated that exactly one real value of $t$ satisfies $t = \sqrt[3]{(t + 0.8)}$. By calculation, verify that this root is between $1.2$ and $1.3$.
(iii)[3]
Apply the iterative rule $t_{n+1} = \sqrt[3]{(t_n + 0.8)}$ to determine $t$ correct to 3 decimal places, writing each iterate to 5 decimal places.
(iv)[3]
Using the $t$-values found in earlier parts, solve $2\tan 2x + 5\tan^2 x = 0$ for $-\pi \le x \le \pi$.