Question

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. $2i\sin n\alpha$
• D. $2\cos n\alpha$

Answer 1

Answer: (D)

$2\cos n\alpha$

Answer 2

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