What is the value of `cot15^@cot20^@cot70^@cot75^@`

Options

A. -1
B. 0
C. 1
D. 2

Answer

To find the value of \( \cot 15^\circ \cdot \cot 20^\circ \cdot \cot 70^\circ \cdot \cot 75^\circ \), we can use the property of cotangent that states:

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

This means that:

\[ \cot 70^\circ = \tan 20^\circ \quad \text{and} \quad \cot 75^\circ = \tan 15^\circ \]

### Step-by-step Solution:

1. **Rewrite cotangent using the complementary angle property**:
\[ \cot 70^\circ = \tan 20^\circ \quad \text{and} \quad \cot 75^\circ = \tan 15^\circ \]

2. **Substitute these values into the original expression**:
\[ \cot 15^\circ \cdot \cot 20^\circ \cdot \tan 20^\circ \cdot \tan 15^\circ \]

3. **Notice that \( \cot \theta \) and \( \tan \theta \) are reciprocals**:
\[ \cot 15^\circ \cdot \tan 15^\circ = 1 \quad \text{and} \quad \cot 20^\circ \cdot \tan 20^\circ = 1 \]

4. **Combine these results**:
\[ 1 \cdot 1 = 1 \]

Thus, the value of \( \cot 15^\circ \cdot \cot 20^\circ \cdot \cot 70^\circ \cdot \cot 75^\circ \) is:

\[ \boxed{1} \]