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 follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ \tan \frac{\pi}{20} \tan \frac{3\pi}{20} \tan \frac{5\pi}{20} \tan \frac{7\pi}{20} \tan \frac{9\pi}{20} \]

### Step 2: Use the identity for complementary angles
We know that:
\[ \tan\left(\frac{\pi}{2} - \theta\right) = \cot \theta \]
We can express \( \tan \frac{7\pi}{20} \) and \( \tan \frac{9\pi}{20} \) as:
\[ \tan \frac{7\pi}{20} = \tan\left(\frac{\pi}{2} - \frac{3\pi}{20}\right) = \cot \frac{3\pi}{20} \]
\[ \tan \frac{9\pi}{20} = \tan\left(\frac{\pi}{2} - \frac{\pi}{20}\right) = \cot \frac{\pi}{20} \]

### Step 3: Substitute the complementary angles
Now, substituting these values into the expression, we get:
\[ \tan \frac{\pi}{20} \tan \frac{3\pi}{20} \tan \frac{5\pi}{20} \cot \frac{3\pi}{20} \cot \frac{\pi}{20} \]

### Step 4: Simplify using cotangent identities
Using the identity \( \cot \theta = \frac{1}{\tan \theta} \), we can rewrite the expression:
\[ \tan \frac{\pi}{20} \tan \frac{3\pi}{20} \tan \frac{5\pi}{20} \cdot \frac{1}{\tan \frac{3\pi}{20}} \cdot \frac{1}{\tan \frac{\pi}{20}} \]

### Step 5: Cancel out terms
Now, we can cancel \( \tan \frac{\pi}{20} \) with \( \frac{1}{\tan \frac{\pi}{20}} \) and \( \tan \frac{3\pi}{20} \) with \( \frac{1}{\tan \frac{3\pi}{20}} \):
\[ \tan \frac{5\pi}{20} = \tan \frac{\pi}{4} \]

### Step 6: Evaluate \( \tan \frac{\pi}{4} \)
We know that:
\[ \tan \frac{\pi}{4} = 1 \]

### Final Result
Thus, the value of the original expression is:
\[ 1 \]