If x + 1 x = 2 , then x^n + 1 x^n = - Vidyadip

MCQ

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

A
${2^n}\cos \alpha $
B
${2^n}\cos n\alpha $
C
$2i\,\sin \,n\,\alpha $
✓
$2\cos \,n\alpha $

✓

Answer

Correct option: D.

$2\cos \,n\alpha $

d
(d) We have, $x + \frac{1}{x} = 2\cos \alpha $

${x^2} + \frac{1}{{{x^2}}} + 2 = 4{\cos ^2}\alpha $.

${x^2} + \frac{1}{{{x^2}}} = 4{\cos ^2}\alpha - 2$

${x^2} + \frac{1}{{{x^2}}} = 2(2{\cos ^2}\alpha - 1) = 2\cos 2\alpha $

Similarly ${x^n} + \frac{1}{{{x^n}}} = 2\cos \,n\alpha $.

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