1. **Problem:** Find $\sec\left(\frac{5\pi}{4}\right)$.\n\n2. **Recall the definition:** $\sec \theta = \frac{1}{\cos \theta}$.\n\n3. **Evaluate $\cos\left(\frac{5\pi}{4}\right)$:**\n$\frac{5\pi}{4}$ is in the third quadrant where cosine is negative.\n$\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$.\n\n4. **Calculate $\sec\left(\frac{5\pi}{4}\right)$:**\n$$\sec\left(\frac{5\pi}{4}\right) = \frac{1}{\cos\left(\frac{5\pi}{4}\right)} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{1}{\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}}.$$\n\n5. **Simplify the fraction:**\n$$-\frac{2}{\sqrt{2}} = -\frac{2}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = -\frac{2\sqrt{2}}{2} = -\sqrt{2}.$$\n\n**Final answer:** $\boxed{-\sqrt{2}}$.