The value of tan `15^@ tan 25^@ tan 45^@ tan 65^@ tan 75^@` is

Options

A. 3
B. `sqrt3`
C. `2`
D. 1

Answer

To find the value of \( \tan 15^\circ \tan 25^\circ \tan 45^\circ \tan 65^\circ \tan 75^\circ \), we can use the properties of the tangent function, particularly the complementary angle identities.

### Step-by-Step Solution:

1. **Identify Complementary Angles**:
We know that:
\[ \tan(90^\circ - \theta) = \cot(\theta) \]
Therefore:
\[ \tan 65^\circ = \cot 25^\circ \quad \text{and} \quad \tan 75^\circ = \cot 15^\circ \]

2. **Rewrite the Expression**:
Using the complementary angle identities, we can rewrite the expression:
\[ \tan 15^\circ \tan 25^\circ \tan 45^\circ \tan 65^\circ \tan 75^\circ = \tan 15^\circ \tan 25^\circ \tan 45^\circ \cot 25^\circ \cot 15^\circ \]

3. **Simplify Using Cotangent**:
Recall that \( \cot \theta = \frac{1}{\tan \theta} \):
\[ \tan 15^\circ \cot 15^\circ = 1 \quad \text{and} \quad \tan 25^\circ \cot 25^\circ = 1 \]
Thus, we can simplify the expression:
\[ \tan 15^\circ \tan 25^\circ \cot 25^\circ \cot 15^\circ = 1 \cdot 1 \cdot \tan 45^\circ \]

4. **Evaluate \( \tan 45^\circ \)**:
We know that:
\[ \tan 45^\circ = 1 \]

5. **Final Calculation**:
Therefore, the entire expression simplifies to:
\[ 1 \cdot 1 \cdot 1 = 1 \]

### Conclusion:
The value of \( \tan 15^\circ \tan 25^\circ \tan 45^\circ \tan 65^\circ \tan 75^\circ \) is \( 1 \).