If n  is a positive integer and ${\left(1+i\sqrt{3}\right)}^n+{\left(1-i\sqrt{3}\right)}^n=2^{n+1}\cos \theta ,$ then the value of $\theta$ is 

$\begin{array}{l}{\left(1+i\sqrt{3}\right)}^n+{\left(1-i\sqrt{3}\right)}^n=2^{n+1}\cos \theta {\left(1+i\sqrt{3}\right)}^n={\left\{2\left[\frac{1}{2}+i\frac{\sqrt{3}}{2}\right]\right\}}^n\\ 2^n{\left[\cos \pi /3+i\sin \pi /3\right]}^n=2^n\left[\cos \frac{n\pi }{3}+i\sin \frac{n\pi }{3}\right]{\left(1-i\sqrt{3}\right)}^n\\ =2^n\left[\cos \frac{n\pi }{3}-i\sin \frac{n\pi }{3}\right]{\left(1+i\sqrt{3}\right)}^n+{\left(1-i\sqrt{3}\right)}^n=2^n\left[2\cos \frac{n\pi }{3}\right]=2^{n+1}\cos \frac{n\pi }{3}\\ \theta =\frac{n\pi }{3}\end{array}$