The value of `cot41^(@).cot42^(@).cot43^(@).cot44^(@).cot45^(@).cot46^(@).cot.47^(@).cot48^(@).cot49^(@)`

Options

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

Answer

To find the value of the expression \( \cot 41^\circ \cdot \cot 42^\circ \cdot \cot 43^\circ \cdot \cot 44^\circ \cdot \cot 45^\circ \cdot \cot 46^\circ \cdot \cot 47^\circ \cdot \cot 48^\circ \cdot \cot 49^\circ \), we can use the property of cotangent that states:

\[ \cot A = \frac{1}{\tan A} \]

Additionally, we can use the identity:

\[ \cot A \cdot \cot (90^\circ - A) = 1 \]

### Step-by-Step Solution:

1. **Pairing the Cotangent Values**:
- We can pair the cotangent values as follows:
- \( \cot 41^\circ \) and \( \cot 49^\circ \)
- \( \cot 42^\circ \) and \( \cot 48^\circ \)
- \( \cot 43^\circ \) and \( \cot 47^\circ \)
- \( \cot 44^\circ \) and \( \cot 46^\circ \)
- This gives us:
\[ \cot 41^\circ \cdot \cot 49^\circ, \quad \cot 42^\circ \cdot \cot 48^\circ, \quad \cot 43^\circ \cdot \cot 47^\circ, \quad \cot 44^\circ \cdot \cot 46^\circ \]

2. **Applying the Identity**:
- Using the identity \( \cot A \cdot \cot (90^\circ - A) = 1 \):
- \( \cot 41^\circ \cdot \cot 49^\circ = 1 \)
- \( \cot 42^\circ \cdot \cot 48^\circ = 1 \)
- \( \cot 43^\circ \cdot \cot 47^\circ = 1 \)
- \( \cot 44^\circ \cdot \cot 46^\circ = 1 \)

3. **Calculating the Total Product**:
- Now we can substitute back into the original expression:
\[ \cot 41^\circ \cdot \cot 42^\circ \cdot \cot 43^\circ \cdot \cot 44^\circ \cdot \cot 45^\circ \cdot \cot 46^\circ \cdot \cot 47^\circ \cdot \cot 48^\circ \cdot \cot 49^\circ \]
- This simplifies to:
\[ 1 \cdot 1 \cdot 1 \cdot 1 \cdot \cot 45^\circ \]

4. **Evaluating \( \cot 45^\circ \)**:
- We know that \( \cot 45^\circ = 1 \).

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

### Final Answer:
The value of \( \cot 41^\circ \cdot \cot 42^\circ \cdot \cot 43^\circ \cdot \cot 44^\circ \cdot \cot 45^\circ \cdot \cot 46^\circ \cdot \cot 47^\circ \cdot \cot 48^\circ \cdot \cot 49^\circ \) is \( \boxed{1} \).