10 By using de Moivre's theorem to express \(\sin 5 \theta\) and \(\cos 5 \theta\) in terms of \(\sin \theta\) and \(\cos \theta\), show that $$\tan 5 \theta = \frac { 5 t - 10 t ^ { 3 } + t ^ { 5 } } { 1 - 10 t ^ { 2 } + 5 t ^ { 4 } }$$ where \(t = \tan \theta\). Show that the roots of the equation \(x ^ { 4 } - 10 x ^ { 2 } + 5 = 0\) are \(\tan \left( \frac { 1 } { 5 } n \pi \right)\) for \(n = 1,2,3,4\). By considering the product of the roots of this equation, find the exact value of \(\tan \left( \frac { 1 } { 5 } \pi \right) \tan \left( \frac { 2 } { 5 } \pi \right)\).

$\cos 5\theta = c^5 - 10c^3s^2 + 5cs^4$ | M1A1 | use of de Moivre for $(c + is)^5$
$\sin 5\theta = 5c^4s - 10c^2s^3 + s^5$ | A1

$\tan 5\theta = \frac{t^5 - 10t^3 + 5t}{1 - 10t^2 + 5t^4}$ | M1A1 | intermediate step needed | [5]

$\tan 5\theta = 0 \Rightarrow \theta = \frac{n\pi}{5}$ | M1

$\text{Solutions } \tan\frac{n\pi}{5}$ for $n = 1, 2, 3, 4$ | A1 | justify values of $n$ | [2]

$\text{Roots } \pm\tan\frac{\pi}{5}, \pm\tan\frac{2\pi}{5}$ | B1

$\text{Product of these roots} = 5$ | M1

$\tan\frac{\pi}{5}\tan\frac{2\pi}{5} = \sqrt{5}$ | A1 | | [3]

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10 By using de Moivre's theorem to express $\sin 5 \theta$ and $\cos 5 \theta$ in terms of $\sin \theta$ and $\cos \theta$, show that

$$\tan 5 \theta = \frac { 5 t - 10 t ^ { 3 } + t ^ { 5 } } { 1 - 10 t ^ { 2 } + 5 t ^ { 4 } }$$

where $t = \tan \theta$.

Show that the roots of the equation $x ^ { 4 } - 10 x ^ { 2 } + 5 = 0$ are $\tan \left( \frac { 1 } { 5 } n \pi \right)$ for $n = 1,2,3,4$.

By considering the product of the roots of this equation, find the exact value of $\tan \left( \frac { 1 } { 5 } \pi \right) \tan \left( \frac { 2 } { 5 } \pi \right)$.

\hfill \mbox{\textit{CAIE FP1 2010 Q10 [10]}}