Proof By Induction Trigonometry Questions, Answers and Solutions

  a - answer    s - solution    v - video

Question 1

Prove by mathematical induction for $n\ge1$ and $n\in\mathbb{Z}^+$:

a) $\cos x\times\cos 2x\times\dots\times\cos(2^{n-1}x)=\frac{\sin(2^nx)}{2^n\sin(x)}$a s v

b) $\cos\left(\frac{x}{2}\right) + \cos\left(\frac{3x}{2}\right)+\cdots + \cos\left(\frac{(2n-1)x}{2}\right) = \frac{\sin nx}{2\sin\left(\frac x 2\right)}$a s v

c) $\frac12+\cos x+\cos 2x+\dotsb+\cos nx=\frac{\sin\left(n+\frac12\right)x}{2\sin\frac12x}$a s v

d) $\sin x+\sin 2x+\sin 3x+\dots+\sin nx=\frac{\sin\frac12(n+1)x\sin\frac12nx}{\sin\frac12x}$a s v

Question 2

(Requires knowledge of complex numbers). Prove by mathematical induction: ( where $i=\sqrt{-1}$)

a) $(\cos(x)+i\sin(x))^n=\cos(nx)+i\sin(nx)$ for $n\ge0$ and $n\in\mathbb{Z}^+$a s v

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