Give the exact value of the expression without using a calculator.
\[
\cos \left(\tan ^{-1} \frac{4}{3}-\tan ^{-1}\left(-\frac{8}{15}\right)\right)
\]
\[
\cos \left(\tan ^{-1} \frac{4}{3}-\tan ^{-1}\left(-\frac{8}{15}\right)\right)=
\]
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Step 1 ：Given the expression \(\cos \left(\tan ^{-1} \frac{4}{3}-\tan ^{-1}\left(-\frac{8}{15}\right)\right)\)

Step 2 ：We can use the formula for the cosine of the difference of two angles, which is given by \(\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B)\)

Step 3 ：Here, A and B are the two angles, which in this case are the inverse tangent of 4/3 and the inverse tangent of -8/15 respectively.

Step 4 ：We know that the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Therefore, we can construct right triangles with the given ratios and use the Pythagorean theorem to find the lengths of the hypotenuses.

Step 5 ：For the first triangle, the sides are 4 and 3, so the hypotenuse is 5. Therefore, \(\cos(A) = \frac{3}{5}\) and \(\sin(A) = \frac{4}{5}\)

Step 6 ：For the second triangle, the sides are 8 and 15, so the hypotenuse is 17. Therefore, \(\cos(B) = \frac{15}{17}\) and \(\sin(B) = -\frac{8}{17}\)

Step 7 ：Substitute these values into the formula for the cosine of the difference of two angles, we get \(\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B) = \frac{3}{5} \cdot \frac{15}{17} + \frac{4}{5} \cdot -\frac{8}{17} = \frac{13}{85}\)

Step 8 ：Final Answer: The exact value of the expression \(\cos \left(\tan ^{-1} \frac{4}{3}-\tan ^{-1}\left(-\frac{8}{15}\right)\right)\) is \(\boxed{\frac{13}{85}}\)