Remember the Pythagorean identity, sin^2(x) + cos^2(x) = 1. If you use this to replace sin^2 with an expression containing cos^2, you can factor the expression.
So you had sin^2 (x) + 2 cos (x) - 2 = 0 and you know by the pythagorean identity that
sin^2 (x) = 1 - cos^2 (x), so replace sin^2 in your equation:
(1 - cos^2 (x) ) + 2 cos (x) - 2 = 0 combine 1-2 = -1 ...
- cos^2 (x) + 2 cos (x) - 1 = 0 take the opposite of each term in the equation...
cos^2 (x) - 2 cos (x) + 1 = 0 and this can be factored.
It might be easier to look at if you let y = cos (x)
y^2 -2y +1 = (y - 1)(y - 1) so replace y with cos (x) and you have
(cos (x) - 1 )(cox (x) - 1 ) = 0 by the zero product property, cos (x) - 1 = 0,
so cos (x) = 1, and the angle that has a cosine of 1 is 0,
so x = 0 (plus any multiple of 360 degrees or 2 pi radians)