1. **State the problem:** Solve the equation $$\sec^2 x - 2 \tan^2 x = 0$$ for $x$. 2. **Recall the identity:** We know that $$\sec^2 x = 1 + \tan^2 x$$. 3. **Substitute the identity into the equation:** $$1 + \tan^2 x - 2 \tan^2 x = 0$$ 4. **Simplify the equation:** $$1 - \tan^2 x = 0$$ 5. **Rearrange to isolate $\tan^2 x$:** $$\tan^2 x = 1$$ 6. **Take the square root of both sides:** $$\tan x = \pm 1$$ 7. **Find general solutions for $\tan x = 1$ and $\tan x = -1$:** - For $\tan x = 1$, solutions are $$x = \frac{\pi}{4} + k\pi$$ - For $\tan x = -1$, solutions are $$x = -\frac{\pi}{4} + k\pi$$ where $k$ is any integer. **Final answer:** $$x = \frac{\pi}{4} + k\pi, -\frac{\pi}{4} + k\pi, \quad k \in \mathbb{Z}$$