Question 13 - A-Level Maths

Two of the solutions to the equation $\cos 6 \theta = 0$ are $\theta = \frac { \pi } { 4 }$ and $\theta = \frac { 3 \pi } { 4 }$
Find the other solutions to the equation $\cos 6 \theta = 0$ for $0 \leq \theta \leq \pi$ 13
Use de Moivre's theorem to show that $$\cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$ 13
Use the fact that $\theta = \frac { \pi } { 4 }$ and $\theta = \frac { 3 \pi } { 4 }$ are solutions to the equation $\cos 6 \theta = 0$ to find a factor of $32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$ in the form ( $a \cos ^ { 2 } \theta + b$ ), where $a$ and $b$ are integers.
[0pt] [4 marks]
Hence show that $$\cos \left( \frac { 11 \pi } { 12 } \right) = - \sqrt { \frac { 2 + \sqrt { 3 } } { 4 } }$$ \includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-25_2492_1721_217_150}