<s>[INST] Why we use $2\cos\theta=\sqrt3$ to find $x^{72}+x^{66}$ given that $x+\frac1x=\sqrt3$? $x + \frac{1}{x} = \sqrt 3$  then find the value of $x^{72}+x^{66}$.
My teacher said that in this case, where $x + \frac{1}{x}$ is equal to any value which is less than $2$, like $1$ or $\sqrt{3}$, just put this value in equal of $2 \cos \theta$ and after you will get value of $\theta$ after that divide the value $180$ by that theta value and after getting the result put that result  in equation $x^n + 1=0$,  as "$n$". 
Please explain what is basic concept behind this.
 [/INST] We have to solve the equation $\ x+\frac{1}{x}=a$. This leads to the equation
$$x^2-ax+1=0$$
If $\ 0\le a<2$, the solutions must be complex.
If $u+vi$ is one of the solutions, the other one is $u-vi$. So, the absolute values of the solutions are equal and multiply to $1$. Therefore, every solution of this equation has absolute value $1$.
If the angle that is formed by a solution is a divisor of $180$ , we have $k\cdot\theta=180$. That means, every solution of the equation satisfies $x^k=-1$.
In the given example, the angle is $30°$, that means $k=6$, so every solution satisfies $x^6=-1$
This immediately shows $x^{72}+x^{66}=0$
What remains is : Why do we take $\ 2cos\theta=a$ ?
The sum of the solutions must be $a$, so the real part of both solutions is $u=\frac{a}{2}$. The absolute value of the solutions is $1$, so we have $u=cos(\theta)$ and therefore $a=2cos(\theta)$. 
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