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Question Number 143150 by ZiYangLee last updated on 10/Jun/21

$$\mathrm{If}\alpha \mathrm{and}\beta \mathrm{are}\mathrm{the}\mathrm{roots}\mathrm{of}\mathrm{the}\mathrm{equation}$$$$\left|\begin{array}{cc}x-\cos \theta & -\sin \theta \\ \sin \theta & x-\cos \theta \end{array}\right|,$$$$\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}{\alpha }^n+{\beta }^n,\mathrm{where}n\in \mathbb{N}.$$

Commented by mr W last updated on 10/Jun/21

$$\left|\begin{array}{cc}x-\cos \theta & -\sin \theta \\ \sin \theta & x-\cos \theta \end{array}\right|isnotanequation.$$$${it}^{\prime }sonlyavalue.$$$$maybeyoumean:$$$$\left|\begin{array}{cc}x-\cos \theta & -\sin \theta \\ \sin \theta & x-\cos \theta \end{array}\right|=0$$

Commented by Dwaipayan Shikari last updated on 10/Jun/21

$${(x-cos\theta )}^2+{sin}^2\theta =0$$$$\Rightarrow x=cos\theta \pm isin\theta =e^{\pm i\theta }$$$${\alpha }^n+{\beta }^n=e^{in\theta }+e^{-in\theta }=2cos\left(n\theta \right)$$

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