1. **State the problem:** Solve the equation $$\tan^2(x) - \sin^2(x) = \sin^4(x) \sec^2(x)$$ for $x$. 2. **Recall definitions and identities:** - $\tan(x) = \frac{\sin(x)}{\cos(x)}$ - $\sec(x) = \frac{1}{\cos(x)}$ - Pythagorean identity: $\sin^2(x) + \cos^2(x) = 1$ 3. **Rewrite each term using sine and cosine:** $$\tan^2(x) = \frac{\sin^2(x)}{\cos^2(x)}$$ $$\sec^2(x) = \frac{1}{\cos^2(x)}$$ Substitute into the equation: $$\frac{\sin^2(x)}{\cos^2(x)} - \sin^2(x) = \sin^4(x) \cdot \frac{1}{\cos^2(x)}$$ 4. **Multiply both sides by $\cos^2(x)$ to clear denominators:** $$\cancel{\cos^2(x)} \cdot \frac{\sin^2(x)}{\cancel{\cos^2(x)}} - \sin^2(x) \cos^2(x) = \sin^4(x) \cancel{\cdot \frac{1}{\cos^2(x)}} \cdot \cos^2(x)$$ Simplifies to: $$\sin^2(x) - \sin^2(x) \cos^2(x) = \sin^4(x)$$ 5. **Bring all terms to one side:** $$\sin^2(x) - \sin^2(x) \cos^2(x) - \sin^4(x) = 0$$ 6. **Factor $\sin^2(x)$ from the first two terms:** $$\sin^2(x)(1 - \cos^2(x)) - \sin^4(x) = 0$$ 7. **Use Pythagorean identity $1 - \cos^2(x) = \sin^2(x)$:** $$\sin^2(x) \cdot \sin^2(x) - \sin^4(x) = 0$$ Which is: $$\sin^4(x) - \sin^4(x) = 0$$ 8. **Simplify:** $$0 = 0$$ This means the equation is an identity for all $x$ where the original expressions are defined (i.e., where $\cos(x) \neq 0$ to avoid division by zero). **Final answer:** The equation holds true for all $x$ such that $\cos(x) \neq 0$, i.e., for all $x \neq \frac{\pi}{2} + k\pi$, $k \in \mathbb{Z}$.