The value of `cot10^@ cot20^@ cot60^@ cot 70^@ cot 80^@` is

Options

A. 1
B. `-1`
C. `sqrt3`
D. `1/sqrt3`

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 use some trigonometric identities and properties.

### Step-by-Step Solution:

1. **Identify the cotangent values**:
\[ \cot 60^\circ = \frac{1}{\sqrt{3}} \]

2. **Use the cotangent identity**:
We know that:
\[ \cot(90^\circ - \theta) = \tan \theta \]
Therefore:
\[ \cot 80^\circ = \cot(90^\circ - 10^\circ) = \tan 10^\circ \]
\[ \cot 70^\circ = \cot(90^\circ - 20^\circ) = \tan 20^\circ \]

3. **Rewrite the expression**:
Substitute the values of \( \cot 70^\circ \) and \( \cot 80^\circ \):
\[ \cot 10^\circ \cdot \cot 20^\circ \cdot \cot 60^\circ \cdot \tan 20^\circ \cdot \tan 10^\circ \]

4. **Combine the cotangent and tangent terms**:
We can rewrite \( \cot 10^\circ \) and \( \cot 20^\circ \) in terms of tangent:
\[ \cot 10^\circ = \frac{1}{\tan 10^\circ}, \quad \cot 20^\circ = \frac{1}{\tan 20^\circ} \]
Thus, the expression becomes:
\[ \frac{1}{\tan 10^\circ} \cdot \frac{1}{\tan 20^\circ} \cdot \frac{1}{\sqrt{3}} \cdot \tan 20^\circ \cdot \tan 10^\circ \]

5. **Simplify the expression**:
The \( \tan 10^\circ \) and \( \tan 20^\circ \) terms will cancel out:
\[ = \frac{\tan 10^\circ \cdot \tan 20^\circ}{\tan 10^\circ \cdot \tan 20^\circ \cdot \sqrt{3}} = \frac{1}{\sqrt{3}} \]

6. **Final result**:
Therefore, the value of \( \cot 10^\circ \cdot \cot 20^\circ \cdot \cot 60^\circ \cdot \cot 70^\circ \cdot \cot 80^\circ \) is:
\[ \frac{1}{\sqrt{3}} \]

### Conclusion:
The final answer is:
\[ \frac{1}{\sqrt{3}} \]