| Exam Board | CAIE |
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| Module | FP1 (Further Pure Mathematics 1) |
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| Year | 2014 |
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| Session | June |
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| Topic | Complex numbers 2 |
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7 Use de Moivre's theorem to show that
$$\tan 5 \theta = \frac { 5 t - 10 t ^ { 3 } + t ^ { 5 } } { 1 - 10 t ^ { 2 } + 5 t ^ { 4 } }$$
where \(t = \tan \theta\).
Deduce that the roots of the equation \(t ^ { 4 } - 10 t ^ { 2 } + 5 = 0\) are \(\pm \tan \frac { 1 } { 5 } \pi\) and \(\pm \tan \frac { 2 } { 5 } \pi\).
Hence show that \(\tan \frac { 1 } { 5 } \pi \tan \frac { 2 } { 5 } \pi = \sqrt { } 5\).