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(a) Using De Moivre's theorem show that \tan 2 \theta=\frac{2 \tan \theta}{1-\tan ^{2} \theta} \text { b) i) Use Euler's theorem to show that } \cos \theta=\frac{1}{2}\left(e^{i \theta}+e^{-i \prime}\right) \text { and find a similar } \text { expression for } \sin \theta \text { b) ii) By expressing } \cos \theta \text { and } \sin \theta \text { in terms of } e^{i \theta} \text { and of } e^{-i \theta} \text { show that: } \cos ^{3} \theta \sin ^{2} \theta=-\frac{1}{16}(\cos 5 \theta+\cos 3 \theta-2 \cos \theta) Hence find the exact value of \int_{0}^{\frac{\pi}{3}} \cos ^{3} \theta \sin ^{2} \theta d \theta \text { Write }\left(1-2 e^{i x}\right)\left(1-2 e^{-i x}\right) \text { in the form } a+b \cos x \text {, where } a \text { and } b \text { are integers. }

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