Let use the trigoniometric identity
\(\cos2x = 2\cos^{2}x - 1:\)

\(2\cos^{2}x - 1 - \cos x = 0\)

Factorize: \((\cos x - 1) (2\cos x + 1) = 0\)

Use the zero product property: \(\cos x - 1 = 0\ or\ 2\cos x + 1 = 0\)

Solve both equations to consine: \(\cos x = 1\ or\ \cos x = -\frac{1}{2}\)

\(\cos x = 1\) when \(x = 0\)

\(x = -\frac{1}{2}\) when \(x = \pi - \pi/3 = (2\pi)/3\ or\ x = \pi + \pi/3 = (4\pi)/3\)

Aswer: \(x = 0, (\frac{2\pi}{3}), (4\pi)/3\)

\(2\cos^{2}x - 1 - \cos x = 0\)

Factorize: \((\cos x - 1) (2\cos x + 1) = 0\)

Use the zero product property: \(\cos x - 1 = 0\ or\ 2\cos x + 1 = 0\)

Solve both equations to consine: \(\cos x = 1\ or\ \cos x = -\frac{1}{2}\)

\(\cos x = 1\) when \(x = 0\)

\(x = -\frac{1}{2}\) when \(x = \pi - \pi/3 = (2\pi)/3\ or\ x = \pi + \pi/3 = (4\pi)/3\)

Aswer: \(x = 0, (\frac{2\pi}{3}), (4\pi)/3\)