| Exam Board | CAIE |
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| Module | FP1 (Further Pure Mathematics 1) |
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| Year | 2013 |
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| Session | November |
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| Topic | Proof by induction |
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9 Prove by mathematical induction that, for every positive integer \(n\),
$$( \cos \theta + i \sin \theta ) ^ { n } = \cos n \theta + i \sin n \theta$$
Express \(\sin ^ { 5 } \theta\) in the form \(p \sin 5 \theta + q \sin 3 \theta + r \sin \theta\), where \(p , q\) and \(r\) are rational numbers to be determined.