Question 8 - A-Level Maths

(a) Use de Moivre's theorem to
1. show that
$$\cos 5 \theta \equiv \cos ^ { 5 } \theta - 10 \cos ^ { 3 } \theta \sin ^ { 2 } \theta + 5 \cos \theta \sin ^ { 4 } \theta$$
find an expression for $\sin 5 \theta$ in terms of $\cos \theta$ and $\sin \theta$
(b) Hence show that $$\tan 5 \theta = \frac { t ^ { 5 } - 10 t ^ { 3 } + 5 t } { 5 t ^ { 4 } - 10 t ^ { 2 } + 1 }$$ where $t = \tan \theta$ and $\cos 5 \theta \neq 0$
(c) Hence find a quadratic equation whose roots $\operatorname { are } ^ { 2 } \tan ^ { 2 } \frac { \pi } { 5 }$ and $\tan ^ { 2 } \frac { 2 \pi } { 5 }$ Give your answer in the form $a x ^ { 2 } + b x + c = 0$ where $a , b$ and $c$ are integers to be found.
(d) Deduce that $\tan \frac { \pi } { 5 } \tan \frac { 2 \pi } { 5 } = \sqrt { 5 }$

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