Solving numerically a non-linear equation.

Question

How is the more appropriate numerical method to solve the equation

$$\cos(2\pi x)+\cos \left(\frac{2\pi N}{x}\right)=2,$$ for a given $N$?

Notice that if $N \in \mathbb{Z}$, then $x\mid N$.

Answer 1

Not sure why you need numerical method.

As for real $A,\cos A\le1$

So, we need $\cos(2\pi x)=\cos\dfrac{2\pi N}x=1$

Also, $\cos B=1\implies B=2m\pi$ where $m$ is any integer.