If x + frac{1}{x} = 2cos alpha,then x^n + fra | vedclass

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(D) Given $x + \frac{1}{x} = 2 \cos \alpha$. Let $x = \cos \alpha + i \sin \alpha = e^{i\alpha}$. Then $\frac{1}{x} = \cos \alpha - i \sin \alpha = e^

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$2 \cos n\alpha$

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(D) Given $x + \frac{1}{x} = 2 \cos \alpha$. Let $x = \cos \alpha + i \sin \alpha = e^{i\alpha}$. Then $\frac{1}{x} = \cos \alpha - i \sin \alpha = e^{-i\alpha}$. Thus,$x + \frac{1}{x} = e^{i\alpha} + e^{-i\alpha} = 2 \cos \alpha$. Now,$x^n + \frac{1}{x^n} = (e^{i\alpha})^n + (e^{-i\alpha})^n = e^{in\alpha} + e^{-in\alpha}$. Using Euler's formula,$e^{in\alpha} + e^{-in\alpha} = 2 \cos n\alpha$. Therefore,$x^n + \frac{1}{x^n} = 2 \cos n\alpha$.

If $x + \frac{1}{x} = 2\cos \alpha$,then $x^n + \frac{1}{x^n} = $

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