1. **State the problem:** Solve the equation $\cos(4x) = -1$ for $x$. 2. **Recall the cosine function properties:** The cosine function equals $-1$ at angles of the form $\pi + 2k\pi$, where $k$ is any integer. 3. **Set up the equation:** $$4x = \pi + 2k\pi$$ 4. **Solve for $x$:** $$x = \frac{\pi + 2k\pi}{4} = \frac{\pi}{4} + \frac{2k\pi}{4} = \frac{\pi}{4} + \frac{k\pi}{2}$$ 5. **Interpretation:** The general solution is $$x = \frac{\pi}{4} + \frac{k\pi}{2}, \quad k \in \mathbb{Z}$$ This means $x$ takes values starting at $\frac{\pi}{4}$ and then increments by $\frac{\pi}{2}$ for any integer $k$. **Final answer:** $$x = \frac{\pi}{4} + \frac{k\pi}{2}, \quad k \in \mathbb{Z}$$