Use the fundamental trigonometric equation
[tex]\sin^2(x) + \cos^2(x) = 1 \implies \sin^2(x) = 1-\cos^2(x)[/tex]
To rewrite the equation as
[tex]1-\cos^2(x) = 1-\cos(x)[/tex]
Subtract 1 from both sides:
[tex]-\cos^2(x) = -\cos(x)[/tex]
Add [tex]\cos^2(x)[/tex] to both sides:
[tex]\cos^2(x) - \cos(x) = 0 \iff \cos(x)\left(\cos(x)-1\right)=0[/tex]
A multipication is zero if and only if one of the factors is zero. So, the solutions are given by
[tex]\cos(x)=0 \iff x=\dfrac{\pi}{2}+k\pi,\quad k \in \mathbb{Z}[/tex]
or
[tex]\cos(x)-1=0 \iff \cos(x)=1 \iff x=2k\pi,\quad k \in \mathbb{Z}[/tex]
