(b) (i) Show that 2 sin x cos(2k + 1)x = sin 2(k + 1)x - sin 2kx. (ii) Using the result from part (i), prove by mathematical induction that cos x + cos 3x + cos 5x + ? + cos(2n - 1)x = sin 2nx / 2 sin x for all integers n, n ? 1.
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Base case (n = 1): $$\sin{Zx} = 2\sin{x}\left(\cos{x}\right)$$ This is true, so the base case holds. Inductive step: Assume the statement is true for n = k: $$\sin{Zkx} = 2\sin{x}\left(\cos{x} + \cos{3x} + \cdots + \cos{(2k-1)x}\right)$$ Now, we need to Show more…
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Key Concepts
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