Find the value of `tan. pi/20tan. (3pi)/20tan. (5pi)/20tan. (7pi)/20tan. (9pi)/20`.

Explanation

To find the value of \( \tan \frac{\pi}{20} \tan \frac{3\pi}{20} \tan \frac{5\pi}{20} \tan \frac{7\pi}{20} \tan \frac{9\pi}{20} \), we can use the properties of trigonometric functions and some identities.

### Step-by-Step Solution:

1. **Recognize the Angles**:
The angles involved are:
\[ \frac{\pi}{20}, \frac{3\pi}{20}, \frac{5\pi}{20}, \frac{7\pi}{20}, \frac{9\pi}{20} \]

2. **Use the Identity**:
We can use the identity:
\[ \tan(\frac{\pi}{2} - x) = \cot(x) \]
This means:
\[ \tan \frac{9\pi}{20} = \cot \frac{\pi}{20}, \quad \tan \frac{7\pi}{20} = \cot \frac{3\pi}{20} \]
Therefore, we can rewrite the product:
\[ \tan \frac{\pi}{20} \tan \frac{3\pi}{20} \tan \frac{5\pi}{20} \tan \frac{7\pi}{20} \tan \frac{9\pi}{20} = \tan \frac{\pi}{20} \tan \frac{3\pi}{20} \tan \frac{5\pi}{20} \cot \frac{3\pi}{20} \cot \frac{\pi}{20} \]

3. **Simplify the Expression**:
The terms \( \tan \frac{3\pi}{20} \) and \( \cot \frac{3\pi}{20} \) will cancel out, and similarly for \( \tan \frac{\pi}{20} \) and \( \cot \frac{\pi}{20} \):
\[ \tan \frac{\pi}{20} \cot \frac{\pi}{20} = 1, \quad \tan \frac{3\pi}{20} \cot \frac{3\pi}{20} = 1 \]
Thus, we are left with:
\[ \tan \frac{5\pi}{20} = \tan \frac{\pi}{4} = 1 \]

4. **Final Result**:
Therefore, the value of the original expression is:
\[ \tan \frac{\pi}{20} \tan \frac{3\pi}{20} \tan \frac{5\pi}{20} \tan \frac{7\pi}{20} \tan \frac{9\pi}{20} = 1 \]

### Conclusion:
The value of \( \tan \frac{\pi}{20} \tan \frac{3\pi}{20} \tan \frac{5\pi}{20} \tan \frac{7\pi}{20} \tan \frac{9\pi}{20} \) is \( 1 \).