Question

if x+1/x=2cos π/5 then show,x5+1/x5= -2
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Answer 1

x+x1=2cos10π

x2+1=2xcox10π

x2−2xcos10π+1=0

x=22cos10π±4cos210π−4=22cos10π±2−sin210π=cos10π+isin10π=ei10π

x5+x5

Answer 2

Step-by-step explanation:

Using De Moivre's theorem on complex number we get

$\displaystyle \sf \: {x}^{5} = \cos \frac{5\pi}{5} \pm \: i \sin \frac{\pi}{5}$

$</p><p>\displaystyle \sf \implies \: {x}^{5} = \cos \pi \pm \: i \sin \pi$

$\displaystyle \sf \implies \: {x}^{5} = - 1 \\ </p><p>\displaystyle \sf \implies \: \frac{1}{ {x}^{5} } = - 1$

Thus we get

$\displaystyle \sf \: {x}^{5} + \frac{1}{ {x}^{5} } \\ \displaystyle \sf \: = - 1 - 1 \\ \displaystyle \sf \: = - 2$