If $\mathrm{x}$ satisfies the equation ${\mathrm{x}}^2-2\mathrm{x} \cos \mathrm{\theta }+1=0$ then the value of ${\mathrm{x}}^{\mathrm{n}}+\frac{1}{{\mathrm{x}}^{\mathrm{n}}}$ is

$\begin{array}{rcl}{\mathrm{x}}^2-2\mathrm{x} \mathrm{cos\theta }+1& =& 0\end{array}$

$\begin{array}{rcl}\mathrm{x}& =& \frac{2\mathrm{cos\theta }\pm \sqrt{4{\cos }^2\mathrm{\theta }-4}}{2}\end{array}$

$\begin{array}{rcl}\mathrm{x}& =& \frac{2\mathrm{cos\theta }\pm 2\mathrm{i} \mathrm{sin\theta }}{2}\end{array}$

$\begin{array}{rcl}\mathrm{x}& =& \mathrm{cos\theta }\pm \mathrm{i} \mathrm{sin\theta }\end{array}$

$\begin{array}{rcl}{\mathrm{x}}^{\mathrm{n}}+\frac{1}{{\mathrm{x}}^{\mathrm{n}}}& =& {\left(\mathrm{cos\theta }+\mathrm{i} \mathrm{sin\theta }\right)}^{\mathrm{n}}+\frac{1}{{\left(\mathrm{cos\theta }+\mathrm{i} \mathrm{sin\theta }\right)}^{\mathrm{n}}}\end{array}$

$\begin{array}{rcl}{\mathrm{x}}^{\mathrm{n}}+\frac{1}{{\mathrm{x}}^{\mathrm{n}}}& =& \cos \mathrm{n\theta }+\mathrm{i} \sin \mathrm{n\theta }+\cos \mathrm{n\theta }-\mathrm{i} \sin \mathrm{n\theta }\end{array}$

$\begin{array}{rcl}{\mathrm{x}}^{\mathrm{n}}+\frac{1}{{\mathrm{x}}^{\mathrm{n}}}& =& 2\cos \mathrm{n\theta }\end{array}$