Question

If $x+\frac{1}{x}=2cos\frac{\pi }{10}$, then find $x^5+\frac{1}{x^5}$

Answer 1

Given

$x+\frac{1}{x}=2cos\frac{\pi }{10}$

$x^2+1=2xcox\frac{\pi }{10}$

$x^2-2xcos\frac{\pi }{10}+1=0$

$x=\frac{2cos\frac{\pi }{10}\pm \sqrt{4co{s}^2\frac{\pi }{10}-4}}{2}=\frac{2cos\frac{\pi }{10}\pm 2\sqrt{-si{n}^2\frac{\pi }{10}}}{2}=cos\frac{\pi }{10}+isin\frac{\pi }{10}=e^{i\frac{\pi }{10}}$

$x^5+\frac{1}{x^5}=(e^{i\frac{\pi }{10}}{)}^5+\frac{1}{(e^{i\frac{\pi }{10}}{)}^5}$

$=e^{i\frac{\pi }{2}}+e^{-i\frac{\pi }{2}}$

$=(cos\frac{\pi }{2}+isin\frac{\pi }{2})+\left(cos\right(-\frac{\pi }{2})+isin(-\frac{\pi }{2}\left)\right)$

$=0+i+0-i$

$=0$