Question 7 - A-Level Maths

1. Use de Moivre's Theorem to show that $$\cos 5 \theta = \cos ^ { 5 } \theta - 10 \cos ^ { 3 } \theta \sin ^ { 2 } \theta + 5 \cos \theta \sin ^ { 4 } \theta$$ and find a similar expression for $\sin 5 \theta$.
2. Deduce that $$\tan 5 \theta = \frac { \tan \theta \left( 5 - 10 \tan ^ { 2 } \theta + \tan ^ { 4 } \theta \right) } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta }$$
Explain why $t = \tan \frac { \pi } { 5 }$ is a root of the equation $$t ^ { 4 } - 10 t ^ { 2 } + 5 = 0$$ and write down the three other roots of this equation in trigonometrical form.
(3 marks)
Deduce that $$\tan \frac { \pi } { 5 } \tan \frac { 2 \pi } { 5 } = \sqrt { 5 }$$