Answer
The solution set to this problem is $$\{\pi\}$$
Work Step by Step
$$\cos^2x+2\cos x+1=0$$ over interval $[0,2\pi)$
1) Solve the equation:
$$\cos^2x+2\cos x+1=0$$
$$(\cos x+1)^2=0$$
$$\cos x+1=0$$
$$\cos x=-1$$
2) Apply the inverse function:
$\cos x=-1$: $\cos x\lt0$ means that the angle of $x$ lies either in quadrant II or III. In fact, looking at the unit circle, we can easily figure out at only $x=\pi$, $\cos x=-1$
Therefore, $$x=\pi$$
In other words, the solution set to this problem is $$\{\pi\}$$
