Give the exact value of the expression without using a calculator.
$$\cos (2\arctan \frac{15}{8})$$
$$\cos (2\arctan \frac{15}{8})=$$
(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 $\cos (2\arctan \frac{15}{8})$ is $\boxed{{\displaystyle -\frac{161}{289}}}$.