Answer
$ x=\dfrac{2\pi}{3}, \dfrac{4\pi}{3}$
Work Step by Step
We have $ \cos (x +\pi) -\cos x -1=-1-2 \cos x=0$
After simplifying, we get $ \cos x \cos \pi -\sin x \sin \pi -\cos x -1=-1-2 \cos x=0$
or, $-\cos x -\cos x -1=-1-2 \cos x$
or, $-1=2 \cos x$
or, $\cos x=\dfrac{-1}{2}$
This yields: $ x=\dfrac{2\pi}{3}, \dfrac{4\pi}{3}$
