$\text{ If }{x}^2-2x\cos \theta +1=0\text{ , then the value of }{x}^{2n}-2x^n\cos n\theta +1$, $\mathrm{n}\in \mathrm{N},\text{ is equal to - }$

$x^2-2x\cos \theta +1=0$

$\therefore \mathrm{x}=\frac{2\cos \theta \pm \sqrt{4{\cos }^2\theta -4}}{2}=\cos \theta \pm \mathrm{isin}\theta$

$\text{ Let }\mathrm{x}=\cos \theta +\mathrm{isin}\theta$

$\therefore x^{2n}-2x^n\cos n\theta +1$

$=\cos 2n\theta +i\sin 2n\theta -2(\cos n\theta +i\sin n\theta )\cos n\theta +1$

$=\cos 2n\theta +1-2{\cos }^{2}n\theta +i(\sin 2n\theta -2\sin n\theta \cos n\theta )$

$=0+\mathrm{i}0=0$