Show that $\cos2nx=\cos((2n+1)x)\cos x+\sin((2n+1)x)\sin x$.[2] Hence prove by mathematical i

Question

Show that $\cos2nx=\cos((2n+1)x)\cos x+\sin((2n+1)x)\sin x$.[2] Hence prove by mathematical induction that $\sin x+\sin3x+\sin5x+\cdots+\sin((2n-1)x)=\frac{1-\cos2nx}{2\sin x}$ for all $n\in\mathbb Z^+$, $\sin x\ne0$.[7] Hence solve $\sin x+\sin3x=\cos x$ for $0<x<\pi$.[5]

Accepted Answer

(c) $x=\frac{\pi}{12},\frac{5\pi}{12},\frac{\pi}{2}$.

Mathematical Induction - Proof by Mathematical Induction

Mathematical Induction - Proof by Mathematical Induction

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