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Let w=cosθ+isinθ, for 43π<θ<45π.
Using the binomial theorem, write down the expansion w3.
Using your result from (a), show that
(i) cos3θ=4cos3θ−3cosθ
(ii) sin3θ=3sinθ−4sin3θ
Hence, or otherwise, show that 1−3tan2A3tanA−tan3A≡tan3A.
Using the identity demonstrated in (c), find the exact value of tan3θ, given that sinθ=53.
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