Great Radical Equation

Geometry Level 3

$4x\sqrt{1-x^2} -\sqrt2 = 0$

For $x \in [-1,1]$ , the roots of the equation above can be expressed in the form $\cos\left( \frac 1 n \pi\right)$ and $\sin\left( \frac 1 n \pi\right)$ , where $n$ is a positive integer.