Question 8 - A-Level Maths

Use de Moivre's theorem to prove that $$\tan 5 \theta \equiv \frac { 5 \tan \theta - 10 \tan ^ { 3 } \theta + \tan ^ { 5 } \theta } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta } .$$
Solve the equation $\tan 5 \theta = 1$, for $0 \leqslant \theta < \pi$.
Show that the roots of the equation $$t ^ { 4 } - 4 t ^ { 3 } - 14 t ^ { 2 } - 4 t + 1 = 0$$ may be expressed in the form $\tan \alpha$, stating the exact values of $\alpha$, where $0 \leqslant \alpha < \pi$. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}