1. **State the problem:** Find the values of $A$ and $B$ such that $$\cos\left(\tan^{-1}\left(\frac{13}{84}\right)\right) = \frac{A}{B}.$$ 2. **Recall the formula:** If $\theta = \tan^{-1}(x)$, then $\tan(\theta) = x = \frac{\text{opposite}}{\text{adjacent}}$. We can use the Pythagorean identity to find $\cos(\theta)$: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}.$$ 3. **Identify sides:** Given $\tan(\theta) = \frac{13}{84}$, set opposite side = 13 and adjacent side = 84. 4. **Calculate hypotenuse:** $$\text{hypotenuse} = \sqrt{13^2 + 84^2} = \sqrt{169 + 7056} = \sqrt{7225} = 85.$$ 5. **Calculate cosine:** $$\cos\left(\tan^{-1}\left(\frac{13}{84}\right)\right) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{84}{85}.$$ 6. **Final answer:** $$A = 84, \quad B = 85.$$