The value of `cot 12^(@) cot 38^(@) cot 52^(@) cot 60^(@) cot 78^(@)` is:

Options

A. 1
B. 0
C. `1/sqrt(2)`
D. `1/sqrt(3)`

Explanation

To solve the problem of finding the value of \( \cot 12^\circ \cot 38^\circ \cot 52^\circ \cot 60^\circ \cot 78^\circ \), we can use the properties of cotangent and the complementary angles.

### Step-by-step Solution:

1. **Identify Complementary Angles**:
We know that \( \cot(90^\circ - \theta) = \tan(\theta) \). This property will help us simplify the cotangent terms:
- \( \cot 12^\circ = \cot(90^\circ - 78^\circ) = \tan 78^\circ \)
- \( \cot 38^\circ = \cot(90^\circ - 52^\circ) = \tan 52^\circ \)

Therefore, we can rewrite the expression as:
\[ \cot 12^\circ \cot 38^\circ \cot 52^\circ \cot 60^\circ \cot 78^\circ = \tan 78^\circ \tan 52^\circ \cot 52^\circ \cot 60^\circ \cot 78^\circ \]

2. **Simplify the Expression**:
Now, substituting the values we have:
\[ = \tan 78^\circ \cdot 1 \cdot \cot 60^\circ \cdot \cot 78^\circ \]
We know that \( \cot 60^\circ = \frac{1}{\sqrt{3}} \).

3. **Using the Identity**:
We also know that \( \tan 78^\circ = \frac{1}{\cot 78^\circ} \). Thus, we can rewrite:
\[ = \tan 78^\circ \cdot \cot 78^\circ \cdot \cot 60^\circ \]
Since \( \tan 78^\circ \cdot \cot 78^\circ = 1 \), we have:
\[ = 1 \cdot \cot 60^\circ = \cot 60^\circ \]

4. **Final Calculation**:
Now substituting the value of \( \cot 60^\circ \):
\[ = \frac{1}{\sqrt{3}} \]

Thus, the value of \( \cot 12^\circ \cot 38^\circ \cot 52^\circ \cot 60^\circ \cot 78^\circ \) is \( \frac{1}{\sqrt{3}} \).

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