1. **State the problem:** Solve the trigonometric equation $$\cos 2x - \cos x + 1 = 0$$. 2. **Recall the double-angle formula:** $$\cos 2x = 2\cos^2 x - 1$$. 3. **Substitute the formula into the equation:** $$2\cos^2 x - 1 - \cos x + 1 = 0$$ 4. **Simplify the equation:** $$2\cos^2 x - \cos x = 0$$ 5. **Factor the equation:** $$\cos x (2\cos x - 1) = 0$$ 6. **Set each factor equal to zero:** - $$\cos x = 0$$ - $$2\cos x - 1 = 0 \Rightarrow \cos x = \frac{1}{2}$$ 7. **Solve for $$x$$:** - For $$\cos x = 0$$, $$x = \frac{\pi}{2} + k\pi, k \in \mathbb{Z}$$ - For $$\cos x = \frac{1}{2}$$, $$x = \pm \frac{\pi}{3} + 2k\pi, k \in \mathbb{Z}$$ **Final answer:** $$x = \frac{\pi}{2} + k\pi \quad \text{or} \quad x = \pm \frac{\pi}{3} + 2k\pi, \quad k \in \mathbb{Z}$$