The value of tan `7^(@) tan 23^(@) tan39^(@) tan 60^(@) tan 51^(@) tan 67^(@) tan 83^(@)` is

Options

A. 0
B. 1
C. `sqrt(3)`
D. `(1)/(sqrt(3))`

Answer

To find the value of the expression \( \tan 7^\circ \tan 23^\circ \tan 39^\circ \tan 60^\circ \tan 51^\circ \tan 67^\circ \tan 83^\circ \), we can utilize the property of complementary angles in trigonometry.

### Step-by-Step Solution:

1. **Identify Complementary Angles**:
- The angles can be paired as follows:
- \( 7^\circ \) and \( 83^\circ \) (since \( 7 + 83 = 90^\circ \))
- \( 23^\circ \) and \( 67^\circ \) (since \( 23 + 67 = 90^\circ \))
- \( 39^\circ \) and \( 51^\circ \) (since \( 39 + 51 = 90^\circ \))
- The angle \( 60^\circ \) stands alone.

2. **Use the Property of Tangent**:
- The property of tangent for complementary angles states that:
\[ \tan(90^\circ - \theta) = \cot(\theta) \]
- Therefore, we can rewrite the product using this property:
\[ \tan 7^\circ = \cot 83^\circ, \quad \tan 23^\circ = \cot 67^\circ, \quad \tan 39^\circ = \cot 51^\circ \]

3. **Rewrite the Expression**:
- The expression can now be rewritten as:
\[ \tan 7^\circ \tan 83^\circ \tan 23^\circ \tan 67^\circ \tan 39^\circ \tan 51^\circ \tan 60^\circ \]
- This gives us:
\[ \tan 7^\circ \cdot \cot 83^\circ \cdot \tan 23^\circ \cdot \cot 67^\circ \cdot \tan 39^\circ \cdot \cot 51^\circ \cdot \tan 60^\circ \]

4. **Simplify the Expression**:
- Since \( \tan \theta \cdot \cot \theta = 1 \), we have:
\[ \tan 7^\circ \cdot \cot 83^\circ = 1, \quad \tan 23^\circ \cdot \cot 67^\circ = 1, \quad \tan 39^\circ \cdot \cot 51^\circ = 1 \]
- Thus, the entire expression simplifies to:
\[ 1 \cdot 1 \cdot 1 \cdot \tan 60^\circ \]

5. **Calculate \( \tan 60^\circ \)**:
- We know that:
\[ \tan 60^\circ = \sqrt{3} \]

6. **Final Result**:
- Therefore, the value of the original expression is:
\[ \tan 7^\circ \tan 23^\circ \tan 39^\circ \tan 60^\circ \tan 51^\circ \tan 67^\circ \tan 83^\circ = \sqrt{3} \]

### Conclusion:
The final answer is \( \sqrt{3} \).