Question 8 - A-Level Maths

Use de Moivre's theorem to find an expression for \(\tan 4 \theta\) in terms of \(\tan \theta\).
Deduce that \(\cot 4 \theta = \frac { \cot ^ { 4 } \theta - 6 \cot ^ { 2 } \theta + 1 } { 4 \cot ^ { 3 } \theta - 4 \cot \theta }\).
Hence show that one of the roots of the equation \(x ^ { 2 } - 6 x + 1 = 0\) is \(\cot ^ { 2 } \left( \frac { 1 } { 8 } \pi \right)\).
Hence find the value of \(\operatorname { cosec } ^ { 2 } \left( \frac { 1 } { 8 } \pi \right) + \operatorname { cosec } ^ { 2 } \left( \frac { 3 } { 8 } \pi \right)\), justifying your answer.