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Question Number 191610 by mehdee42 last updated on 27/Apr/23

$$Q:ifx+\frac{1}{x}=2cos\left(\theta \right)proveit{x}^n+\frac{1}{x^n}=2cos\left(n\theta \right)$$

Answered by Frix last updated on 27/Apr/23

$$z+\frac{1}{z}=2c\Rightarrow z=c\pm \sqrt{c^2-1}\Rightarrow$$$$x=\cos \theta \pm \mathrm{i}\sin \theta ={\mathrm{e}}^{\pm \mathrm{i}\theta }$$$$x^n+\frac{1}{x^n}={\mathrm{e}}^{\pm \mathrm{i}n\theta }+{\mathrm{e}}^{\mp \mathrm{i}n\theta }=2\cos n\theta$$

Commented by mehdee42 last updated on 27/Apr/23

$$verynicesir$$

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