IAL 2021 June Q7

(a) Use de Moivre’s theorem to show that

\begin{align*} \tan 4\theta = \frac{4\tan\theta - 4\tan^3\theta}{1 - 6\tan^2\theta + \tan^4\theta} \end{align*}

(6)

(b) Use the identity given in part (a) to find the $2$ positive roots of

\begin{align*} x^4 + 2x^3 - 6x^2 - 2x + 1 = 0 \end{align*}

giving your answers to $3$ significant figures.

(3)