Give the exact value of the expression without using a calculator.
\[
\cos \left(2 \arctan \frac{15}{8}\right)
\]
\[
\cos \left(2 \arctan \frac{15}{8}\right)=
\]
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Step 1 ：The given expression is in the form of cosine of double angle. We can use the identity of cosine of double angle to simplify the expression. The identity is: \(\cos(2A) = 1 - 2\sin^2(A)\) where A is the angle. In this case, A is \(\arctan \frac{15}{8}\).

Step 2 ：We can also use the identity of tangent to find the value of \(\sin(A)\). The identity is: \(\tan(A) = \frac{\sin(A)}{\cos(A)}\). From this, we can find the value of \(\sin(A)\) and substitute it into the identity of cosine of double angle to find the value of the given expression.

Step 3 ：We know that \(\tan(A) = \frac{\sin(A)}{\cos(A)}\), so we can express \(\sin(A)\) and \(\cos(A)\) in terms of \(\tan(A)\).

Step 4 ：We also know that \(\sin^2(A) + \cos^2(A) = 1\), so we can use this identity to express \(\sin(A)\) and \(\cos(A)\) in terms of \(\tan(A)\).

Step 5 ：Then we can substitute these expressions into the identity of cosine of double angle to find the exact value of the expression.

Step 6 ：The exact value of the expression \(\cos \left(2 \arctan \frac{15}{8}\right)\) is \(\boxed{-\frac{161}{289}}\).