1. **State the problem:** We need to find the value of $$\cos\left(\tan^{-1}\left(\frac{8}{15}\right) + \tan^{-1}\left(\frac{15}{8}\right)\right)$$ where both angles are from the right triangle with sides 8, 15, and hypotenuse 17. 2. **Recall the formula for cosine of sum of two angles:** $$\cos(A+B) = \cos A \cos B - \sin A \sin B$$ 3. **Identify angles:** Let $$A = \tan^{-1}\left(\frac{8}{15}\right)$$ and $$B = \tan^{-1}\left(\frac{15}{8}\right)$$. 4. **Find $$\sin$$ and $$\cos$$ for each angle using the triangle sides:** - For angle $$A$$, opposite side = 8, adjacent side = 15, hypotenuse = 17. $$\sin A = \frac{8}{17}, \quad \cos A = \frac{15}{17}$$ - For angle $$B$$, opposite side = 15, adjacent side = 8, hypotenuse = 17. $$\sin B = \frac{15}{17}, \quad \cos B = \frac{8}{17}$$ 5. **Apply the cosine sum formula:** $$\cos(A+B) = \cos A \cos B - \sin A \sin B = \frac{15}{17} \times \frac{8}{17} - \frac{8}{17} \times \frac{15}{17}$$ 6. **Calculate:** $$\cos(A+B) = \frac{120}{289} - \frac{120}{289} = 0$$ 7. **Interpretation:** The cosine of the sum of these two angles is 0, which means the sum of the angles is $$90^\circ$$ or $$\frac{\pi}{2}$$ radians, consistent with the fact that these two angles are complementary in the right triangle. **Final answer:** $$\boxed{0}$$