1. **Stating the problem:** We are given the complex number with argument $\text{Arg } Z = -\pi$ and real part $\text{Re } Z = -1$, and we want to evaluate $\cos^4 x$ for some $x$. 2. **Understanding the complex number:** The argument $\text{Arg } Z = -\pi$ means the angle of the complex number $Z$ in the complex plane is $-\pi$ radians, which points directly to the negative real axis. 3. **Using the real part:** Since $\text{Re } Z = -1$ and the argument is $-\pi$, the complex number is $Z = -1 + 0i$. 4. **About $\cos^4 x$:** The expression $\cos^4 x$ means $(\cos x)^4$, the cosine of $x$ raised to the fourth power. 5. **No specific $x$ given:** Since $x$ is not specified, we cannot compute a numeric value for $\cos^4 x$. However, we can express $\cos^4 x$ in terms of multiple angles using the power-reduction formula. 6. **Power-reduction formula:** Recall that $$\cos^2 x = \frac{1 + \cos 2x}{2}$$ Therefore, $$\cos^4 x = (\cos^2 x)^2 = \left(\frac{1 + \cos 2x}{2}\right)^2 = \frac{1}{4}(1 + 2\cos 2x + \cos^2 2x)$$ 7. **Reduce $\cos^2 2x$ further:** Using the formula again, $$\cos^2 2x = \frac{1 + \cos 4x}{2}$$ 8. **Substitute back:** $$\cos^4 x = \frac{1}{4}\left(1 + 2\cos 2x + \frac{1 + \cos 4x}{2}\right) = \frac{1}{4}\left(1 + 2\cos 2x + \frac{1}{2} + \frac{\cos 4x}{2}\right)$$ 9. **Simplify:** $$\cos^4 x = \frac{1}{4}\left(\frac{3}{2} + 2\cos 2x + \frac{\cos 4x}{2}\right) = \frac{3}{8} + \frac{1}{2}\cos 2x + \frac{1}{8}\cos 4x$$ **Final expression:** $$\boxed{\cos^4 x = \frac{3}{8} + \frac{1}{2}\cos 2x + \frac{1}{8}\cos 4x}$$ This formula expresses $\cos^4 x$ in terms of cosines of multiple angles, which is useful for integration or simplification. --- **Summary:** - The complex number $Z$ is $-1$ on the real axis. - The expression $\cos^4 x$ can be rewritten using power-reduction formulas as above.