Step-by-step Solution:

\begin{aligned} &\text{Given: } \cos \theta = \frac{4}{5}, \quad \cos \phi = \frac{12}{13} \\ &\text{Both } \theta \text{ and } \phi \text{ are in the fourth quadrant.} \\ \\ &\text{Step 1: Find } \sin \theta \text{ and } \sin \phi. \\ &\text{Using the Pythagorean identity: } \sin^2 x + \cos^2 x = 1, \text{ we get} \\ \\ &\sin \theta = -\sqrt{1 - \cos^2 \theta} = -\sqrt{1 - \left(\frac{4}{5}\right)^2} \\ &= -\sqrt{1 - \frac{16}{25}} = -\sqrt{\frac{9}{25}} = -\frac{3}{5} \\ \\ &\sin \phi = -\sqrt{1 - \cos^2 \phi} = -\sqrt{1 - \left(\frac{12}{13}\right)^2} \\ &= -\sqrt{1 - \frac{144}{169}} = -\sqrt{\frac{25}{169}} = -\frac{5}{13} \\ \\ &\text{Step 2: Use the cosine addition formula.} \\ &\cos(\theta + \phi) = \cos \theta \cos \phi - \sin \theta \sin \phi \\ \\ &\cos(\theta + \phi) = \left(\frac{4}{5} \times \frac{12}{13}\right) - \left(-\frac{3}{5} \times -\frac{5}{13}\right) \\ &= \left(\frac{48}{65}\right) - \left(\frac{15}{65}\right) \\ &= \frac{48 - 15}{65} = \frac{33}{65} \\ \\ &\text{Final Answer: } \cos(\theta + \phi) = \frac{33}{65} \end{aligned}