To solve the equation [tex]\(\cos x - 1 = -\cos x\)[/tex] on the interval [tex]\([0, 2\pi)\)[/tex], we follow these steps:
1. Simplify the equation:
[tex]\[\cos x - 1 = -\cos x\][/tex]
2. Combine like terms:
[tex]\[\cos x + \cos x - 1 = 0\][/tex]
[tex]\[2\cos x - 1 = 0\][/tex]
3. Isolate [tex]\(\cos x\)[/tex]:
[tex]\[2\cos x = 1\][/tex]
[tex]\[\cos x = \frac{1}{2}\][/tex]
4. Determine the values of [tex]\(x\)[/tex] that satisfy [tex]\(\cos x = \frac{1}{2}\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex]:
- The cosine function [tex]\(\cos x\)[/tex] equals [tex]\(\frac{1}{2}\)[/tex] at:
[tex]\[x = \frac{\pi}{3}\][/tex]
[tex]\[x = 5\pi/3\][/tex]
5. We explicitly state the solutions found within the given interval [tex]\([0, 2\pi)\)[/tex]:
[tex]\[
x = \frac{\pi}{3}, \; \frac{5\pi}{3}
\][/tex]
Thus, the solutions to the equation [tex]\(\cos x - 1 = -\cos x\)[/tex] on the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
\boxed{\frac{\pi}{3}, \frac{5\pi}{3}}
\][/tex]
