Solve the equation $\sqrt{1-x}=2x^2-1+2x\sqrt{1-x^2}$ Solve the following equation: $\sqrt{1-x}=2x^2-1+2x\sqrt{1-x^2}$
Unfortunately I have no idea.

Put $x=\cos \theta$, then:
$\sqrt{1-\cos \theta} = 2\cos^2 \theta -1 \pm 2\cos \theta\sin\theta = \cos 2\theta \pm \sin 2\theta$.
Squaring ,we get:
$1-\cos \theta = 1 \pm \sin 4\theta$.
Therefore $\cos \theta = \pm\sin 4\theta$. The solutions of this equation are easy to find. Also, one must eliminate extraneous solutions e.g. $126^\circ$.
