Show that $\sin 2 nx = \sin((2n + 1)x) \cos x - \cos((2n + 1)x) \sin x$.[2] Hence prove, by i

Question

Show that $\sin 2 nx = \sin((2n + 1)x) \cos x - \cos((2n + 1)x) \sin x$.[2] Hence prove, by induction, that $$\cos x + \cos 3x + \cos 5x + ... + \cos((2n - 1)x) = \frac{\sin 2nx}{2 \sin x},$$ for all $n \in \mathbb{Z}^+$, $\sin x \neq 0$. [12]

Accepted Answer

(b) $\cos x+\cos3x+\cdots+\cos((2n-1)x)=\dfrac{\sin2nx}{2\sin x}$

Mathematical Induction

Mathematical Induction

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