7. It is given that there is exactly one value of \(x\), where \(0 < x < \pi\), that satisfies the equation
$$3 \tan 2 x - 8 \tan x = 4$$
- Show that \(t = \sqrt [ 3 ] { \frac { 1 } { 2 } + \frac { 1 } { 4 } t - \frac { 1 } { 2 } t ^ { 2 } }\), where \(t = \tan x\).
- Show by calculation that the value of \(t\) satisfying the equation in part (i) lies between 0.7 and 0.8 .
- Use an iterative process based on the equation in part (i) to find the value of \(t\) correct to 4 significant figures. Use a starting value of 0.75 and show the result of each iteration.
- Solve the equation \(3 \tan 4 y - 8 \tan 2 y = 4\) for \(0 < y < \frac { 1 } { 2 } \pi\).
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