1. **State the problem:** Solve the equation $2\cos^2 x = 1$ for $0 < x < 360$ degrees. 2. **Rewrite the equation:** Divide both sides by 2 to isolate $\cos^2 x$: $$\cos^2 x = \frac{1}{2}$$ 3. **Take the square root:** Recall that $\cos^2 x = (\cos x)^2$, so $$\cos x = \pm \sqrt{\frac{1}{2}} = \pm \frac{\sqrt{2}}{2}$$ 4. **Find angles where $\cos x = \frac{\sqrt{2}}{2}$:** This occurs at $$x = 45^\circ, 315^\circ$$ 5. **Find angles where $\cos x = -\frac{\sqrt{2}}{2}$:** This occurs at $$x = 135^\circ, 225^\circ$$ 6. **Final answer:** The solutions in the interval $0 < x < 360$ are $$x = 45^\circ, 135^\circ, 225^\circ, 315^\circ$$