1. **State the problem:** Solve the trigonometric equation $$\sin^2 x = 3 \cos^2 x$$ for $x$. 2. **Use the Pythagorean identity:** Recall that $$\sin^2 x + \cos^2 x = 1$$. 3. **Rewrite the equation:** Substitute $$\sin^2 x = 1 - \cos^2 x$$ into the original equation: $$1 - \cos^2 x = 3 \cos^2 x$$ 4. **Simplify the equation:** $$1 = 3 \cos^2 x + \cos^2 x = 4 \cos^2 x$$ 5. **Isolate $$\cos^2 x$$:** $$\cos^2 x = \frac{1}{4}$$ 6. **Take the square root:** $$\cos x = \pm \frac{1}{2}$$ 7. **Find general solutions:** - For $$\cos x = \frac{1}{2}$$, solutions are $$x = 2k\pi \pm \frac{\pi}{3}$$. - For $$\cos x = -\frac{1}{2}$$, solutions are $$x = 2k\pi \pm \frac{2\pi}{3}$$. 8. **Final answer:** $$x = 2k\pi \pm \frac{\pi}{3} \quad \text{or} \quad x = 2k\pi \pm \frac{2\pi}{3}, \quad k \in \mathbb{Z}$$