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 \( \boxed{-\sqrt{2 + \sqrt{2}}} \)