Use a half-angle formula to find the exact value of $\tan \frac{5\pi }{8}$.

Step 1 ：First, we need to calculate the cosine of $\frac{5\pi }{4}$, then substitute it into the half-angle formula for tangent.

Step 2 ：Let $x=\frac{5\pi }{4}$

Step 3 ：Calculate $\cos (x)=-0.7071067811865477$

Step 4 ：Substitute $\cos (x)$ into the half-angle formula for tangent to get $\tan (\frac{x}{2})=\sqrt{\frac{1-\cos (x)}{1+\cos (x)}}=2.414213562373096$

Step 5 ：However, we need to consider the sign of the result. The tangent function is positive in the first and third quadrants, and $\frac{5\pi }{8}$ lies in the second quadrant. Therefore, the exact value of $\tan \frac{5\pi }{8}$ should be negative.

Step 6 ：Final Answer: The exact value of $\tan \frac{5\pi }{8}$ is $-\sqrt{2+\sqrt{2}}$ or approximately -2.414213562373096. So, the final answer is $\boxed{{\displaystyle -\sqrt{2+\sqrt{2}}}}$