Question

sin^2(x) - cos^2(x) = 1 + cos(x)

the answer is equal to or greater than 0 and less than 2 pi

Answer 1

Rewrite as
1 - 2 cos^2 x = 1 + cos x.

-2 cos^2 x = cos x

cosx (1 + 2 cosx) = 0

cosx = 0 or -1/2

There are 4 possible values of x between 0 and 2 pi. One is pi/2 and another is 2 pi/3. Try for the other two.