Find the value of ` tan 4^@ tan 43^@ tan 47 ^@ tan 86 ^@`

Options

A. `2/3`
B. `1`
C. `1/2`
D. `2`

Answer

To find the value of \( \tan 4^\circ \tan 43^\circ \tan 47^\circ \tan 86^\circ \), we can use some trigonometric identities and properties.

### Step-by-step Solution:

1. **Identify the angles**: We have the angles \( 4^\circ, 43^\circ, 47^\circ, \) and \( 86^\circ \).

2. **Use the identity \( \tan(90^\circ - \theta) = \cot(\theta) \)**:
- Notice that \( \tan 47^\circ = \tan(90^\circ - 43^\circ) = \cot 43^\circ \)
- Also, \( \tan 86^\circ = \tan(90^\circ - 4^\circ) = \cot 4^\circ \)

3. **Rewrite the expression**:
\[ \tan 4^\circ \tan 43^\circ \tan 47^\circ \tan 86^\circ = \tan 4^\circ \tan 43^\circ \cot 43^\circ \cot 4^\circ \]

4. **Apply the cotangent identity**:
- Since \( \tan \theta \cot \theta = 1 \), we can simplify:
\[ \tan 4^\circ \cot 4^\circ = 1 \quad \text{and} \quad \tan 43^\circ \cot 43^\circ = 1 \]

5. **Combine the results**:
\[ \tan 4^\circ \tan 43^\circ \cot 43^\circ \cot 4^\circ = 1 \cdot 1 = 1 \]

Thus, the value of \( \tan 4^\circ \tan 43^\circ \tan 47^\circ \tan 86^\circ \) is \( \boxed{1} \).