The value of `cot 5^(@).cot 10^(@).cot 15^(@)……cot 85^(@)` is

Options

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

Answer

To find the value of \( \cot 5^\circ \cdot \cot 10^\circ \cdot \cot 15^\circ \cdots \cot 85^\circ \), we can use the property of cotangent that relates angles that sum to \( 90^\circ \).

### Step-by-step Solution:

1. **Pairing the Cotangent Terms**:
We can pair the cotangent terms as follows:
\[ \cot 5^\circ \cdot \cot 85^\circ, \quad \cot 10^\circ \cdot \cot 80^\circ, \quad \cot 15^\circ \cdot \cot 75^\circ, \quad \ldots, \quad \cot 45^\circ \]

2. **Using the Cotangent Identity**:
We know that:
\[ \cot(90^\circ - \theta) = \tan(\theta) \]
Therefore, we can rewrite the pairs:
\[ \cot 5^\circ \cdot \cot 85^\circ = \cot 5^\circ \cdot \tan 5^\circ = 1 \]
\[ \cot 10^\circ \cdot \cot 80^\circ = \cot 10^\circ \cdot \tan 10^\circ = 1 \]
\[ \cot 15^\circ \cdot \cot 75^\circ = \cot 15^\circ \cdot \tan 15^\circ = 1 \]
\[ \cot 20^\circ \cdot \cot 70^\circ = \cot 20^\circ \cdot \tan 20^\circ = 1 \]
\[ \cot 25^\circ \cdot \cot 65^\circ = \cot 25^\circ \cdot \tan 25^\circ = 1 \]
\[ \cot 30^\circ \cdot \cot 60^\circ = \cot 30^\circ \cdot \tan 30^\circ = 1 \]
\[ \cot 35^\circ \cdot \cot 55^\circ = \cot 35^\circ \cdot \tan 35^\circ = 1 \]
\[ \cot 40^\circ \cdot \cot 50^\circ = \cot 40^\circ \cdot \tan 40^\circ = 1 \]
\[ \cot 45^\circ = 1 \]

3. **Combining the Results**:
Since each pair of cotangent terms multiplies to 1, we can conclude:
\[ \cot 5^\circ \cdot \cot 85^\circ = 1, \quad \cot 10^\circ \cdot \cot 80^\circ = 1, \quad \ldots, \quad \cot 45^\circ = 1 \]
Therefore, the entire product simplifies to:
\[ 1 \cdot 1 \cdot 1 \cdots \cdot 1 = 1 \]

### Final Answer:
Thus, the value of \( \cot 5^\circ \cdot \cot 10^\circ \cdot \cot 15^\circ \cdots \cot 85^\circ \) is \( \boxed{1} \).