1. **Problem Statement:** Find the exact value of $\cos\left(\frac{13\pi}{12}\right)$.\n\n2. **Formula and Rules:** We use the cosine addition formula: $$\cos(a+b) = \cos a \cos b - \sin a \sin b.$$\n\n3. **Step 1: Express $\frac{13\pi}{12}$ as a sum of angles with known cosine and sine values.**\nNote that $\frac{13\pi}{12} = \pi + \frac{\pi}{12}$.\n\n4. **Step 2: Use the cosine addition formula with $a=\pi$ and $b=\frac{\pi}{12}$.**\n$$\cos\left(\pi + \frac{\pi}{12}\right) = \cos \pi \cos \frac{\pi}{12} - \sin \pi \sin \frac{\pi}{12}.$$\n\n5. **Step 3: Substitute known values:**\n$\cos \pi = -1$, $\sin \pi = 0$.\nSo, $$\cos\left(\frac{13\pi}{12}\right) = (-1) \cdot \cos \frac{\pi}{12} - 0 \cdot \sin \frac{\pi}{12} = -\cos \frac{\pi}{12}.$$\n\n6. **Step 4: Find $\cos \frac{\pi}{12}$.**\nUse the half-angle formula or express $\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$. Using the cosine difference formula: $$\cos(a-b) = \cos a \cos b + \sin a \sin b,$$ with $a=\frac{\pi}{3}$ and $b=\frac{\pi}{4}$.\n\n7. **Step 5: Calculate:**\n$$\cos \frac{\pi}{12} = \cos \frac{\pi}{3} \cos \frac{\pi}{4} + \sin \frac{\pi}{3} \sin \frac{\pi}{4} = \left(\frac{1}{2}\right) \left(\frac{\sqrt{2}}{2}\right) + \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}.$$\n\n8. **Step 6: Substitute back:**\n$$\cos\left(\frac{13\pi}{12}\right) = -\frac{\sqrt{6} + \sqrt{2}}{4}.$$\n\n**Final answer:** $$\boxed{-\frac{\sqrt{6} + \sqrt{2}}{4}}.$$