Find the value of `cot25° × cot35° × cot45° × cot55° × cot65°`

Options

A. `√3`
B. `2`
C. `1`
D. `0`

Answer

To find the value of \( \cot 25^\circ \times \cot 35^\circ \times \cot 45^\circ \times \cot 55^\circ \times \cot 65^\circ \), we can use the property of cotangent that states:

\[ \cot \theta \times \cot (90^\circ - \theta) = 1 \]

### Step-by-step Solution:

1. **Identify pairs using the cotangent property**:
- We can pair \( \cot 25^\circ \) with \( \cot 65^\circ \) because:
\[ \cot 25^\circ = \cot (90^\circ - 65^\circ) \]
- We can pair \( \cot 35^\circ \) with \( \cot 55^\circ \) because:
\[ \cot 35^\circ = \cot (90^\circ - 55^\circ) \]

2. **Rewrite the expression**:
- The original expression can be rewritten as:
\[ \cot 25^\circ \times \cot 65^\circ \times \cot 35^\circ \times \cot 55^\circ \times \cot 45^\circ \]

3. **Apply the cotangent property**:
- Using the property:
\[ \cot 25^\circ \times \cot 65^\circ = 1 \]
\[ \cot 35^\circ \times \cot 55^\circ = 1 \]

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

5. **Combine the results**:
- Now, substituting back into the expression:
\[ (\cot 25^\circ \times \cot 65^\circ) \times (\cot 35^\circ \times \cot 55^\circ) \times \cot 45^\circ = 1 \times 1 \times 1 = 1 \]

### Final Answer:
Thus, the value of \( \cot 25^\circ \times \cot 35^\circ \times \cot 45^\circ \times \cot 55^\circ \times \cot 65^\circ \) is \( \boxed{1} \).