Question 4 - A-Level Maths

Use de Moivre's theorem to show that $$\cot 6 \theta = \frac { \cot ^ { 6 } \theta - 15 \cot ^ { 4 } \theta + 15 \cot ^ { 2 } \theta - 1 } { 6 \cot ^ { 5 } \theta - 20 \cot ^ { 3 } \theta + 6 \cot \theta } .$$ \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-08_2718_35_107_2012}
\includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-09_2723_33_99_22}
Hence obtain the roots of the equation $$x ^ { 6 } - 6 x ^ { 5 } - 15 x ^ { 4 } + 20 x ^ { 3 } + 15 x ^ { 2 } - 6 x - 1 = 0$$ in the form $\cot ( q \pi )$, where $q$ is a rational number.