Find the value of `cot10^(@).cot20^(@).cot60^(@).cot70^(@).cot80^(@)`.

Options

A. 1
B. `-1`
C. `(1)/(sqrt(3))`
D. `sqrt(3)`

Answer

To find the value of \( \cot(10^\circ) \cdot \cot(20^\circ) \cdot \cot(60^\circ) \cdot \cot(70^\circ) \cdot \cot(80^\circ) \), we can utilize some properties of trigonometric functions.

### Step-by-Step Solution:

1. **Use the cotangent identity**:
We know that \( \cot(90^\circ - \theta) = \tan(\theta) \) and \( \cot(\theta) = \frac{1}{\tan(\theta)} \). Therefore, we can express \( \cot(70^\circ) \) and \( \cot(80^\circ) \) in terms of \( \cot(20^\circ) \) and \( \cot(10^\circ) \):
\[ \cot(70^\circ) = \cot(90^\circ - 20^\circ) = \tan(20^\circ \]
\[ \cot(80^\circ) = \cot(90^\circ - 10^\circ) = \tan(10^\circ) \]

2. **Rewrite the expression**:
Substitute the identities into the original expression:
\[ \cot(10^\circ) \cdot \cot(20^\circ) \cdot \cot(60^\circ) \cdot \tan(20^\circ) \cdot \tan(10^\circ) \]

3. **Combine terms**:
Notice that \( \cot(10^\circ) \cdot \tan(10^\circ) = 1 \) and \( \cot(20^\circ) \cdot \tan(20^\circ) = 1 \):
\[ 1 \cdot 1 \cdot \cot(60^\circ) \]

4. **Evaluate \( \cot(60^\circ) \)**:
We know that \( \cot(60^\circ) = \frac{1}{\sqrt{3}} \):
\[ \cot(60^\circ) = \frac{1}{\sqrt{3}} \]

5. **Final result**:
Therefore, the entire expression simplifies to:
\[ \cot(10^\circ) \cdot \cot(20^\circ) \cdot \cot(60^\circ) \cdot \cot(70^\circ) \cdot \cot(80^\circ) = \frac{1}{\sqrt{3}} \]

### Final Answer:
\[ \frac{1}{\sqrt{3}} \]