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

Options

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

Answer

To find the value of \( \cot 15^\circ \cot 20^\circ \cot 70^\circ \cot 75^\circ \), we can use the properties of cotangent and some trigonometric identities.

### Step-by-Step Solution:

1. **Use the cotangent identity**: Recall that \( \cot(90^\circ - \theta) = \tan(\theta) \). This means:
- \( \cot 75^\circ = \tan 15^\circ \)
- \( \cot 70^\circ = \tan 20^\circ \)

2. **Rewrite the expression**: Substitute the identities into the original expression:
\[ \cot 15^\circ \cot 20^\circ \cot 70^\circ \cot 75^\circ = \cot 15^\circ \cot 20^\circ \tan 20^\circ \tan 15^\circ \]

3. **Simplify the expression**: Notice that \( \cot \theta \tan \theta = 1 \):
\[ \cot 15^\circ \tan 15^\circ = 1 \quad \text{and} \quad \cot 20^\circ \tan 20^\circ = 1 \]
Therefore,
\[ \cot 15^\circ \cot 20^\circ \tan 20^\circ \tan 15^\circ = 1 \cdot 1 = 1 \]

4. **Final result**: Thus, the value of \( \cot 15^\circ \cot 20^\circ \cot 70^\circ \cot 75^\circ \) is:
\[ \boxed{1} \]