`cot(pi/20).cot((3pi)/20)cot((5pi)/20)cot((7pi)/20)cot((9pi)/20)=....`

Options

A. -1
B. 0
C. 1
D. none of these

Answer

To solve the expression \( \cot\left(\frac{\pi}{20}\right) \cdot \cot\left(\frac{3\pi}{20}\right) \cdot \cot\left(\frac{5\pi}{20}\right) \cdot \cot\left(\frac{7\pi}{20}\right) \cdot \cot\left(\frac{9\pi}{20}\right) \), we will use the properties of cotangent and the identities involving complementary angles.

### Step-by-Step Solution:

1. **Identify Complementary Angles**:
We know that:
\[ \cot\left(\frac{\pi}{2} - x\right) = \tan(x) \]
This means that:
\[ \cot\left(\frac{9\pi}{20}\right) = \tan\left(\frac{\pi}{20}\right) \]
and
\[ \cot\left(\frac{7\pi}{20}\right) = \tan\left(\frac{3\pi}{20}\right) \]

2. **Rewrite the Expression**:
Using the complementary angle identities, we can rewrite the expression:
\[ \cot\left(\frac{\pi}{20}\right) \cdot \cot\left(\frac{3\pi}{20}\right) \cdot \cot\left(\frac{5\pi}{20}\right) \cdot \cot\left(\frac{7\pi}{20}\right) \cdot \cot\left(\frac{9\pi}{20}\right) \]
becomes:
\[ \cot\left(\frac{\pi}{20}\right) \cdot \tan\left(\frac{\pi}{20}\right) \cdot \cot\left(\frac{3\pi}{20}\right) \cdot \tan\left(\frac{3\pi}{20}\right) \cdot \cot\left(\frac{5\pi}{20}\right) \]

3. **Simplify Using Identities**:
We know that:
\[ \cot(x) \cdot \tan(x) = 1 \]
Therefore:
\[ \cot\left(\frac{\pi}{20}\right) \cdot \tan\left(\frac{\pi}{20}\right) = 1 \]
and
\[ \cot\left(\frac{3\pi}{20}\right) \cdot \tan\left(\frac{3\pi}{20}\right) = 1 \]
The middle term, \( \cot\left(\frac{5\pi}{20}\right) \), simplifies to:
\[ \cot\left(\frac{\pi}{4}\right) = 1 \]

4. **Final Calculation**:
Thus, the entire expression simplifies to:
\[ 1 \cdot 1 \cdot 1 = 1 \]

### Conclusion:
The value of the expression \( \cot\left(\frac{\pi}{20}\right) \cdot \cot\left(\frac{3\pi}{20}\right) \cdot \cot\left(\frac{5\pi}{20}\right) \cdot \cot\left(\frac{7\pi}{20}\right) \cdot \cot\left(\frac{9\pi}{20}\right) \) is \( \boxed{1} \).