The value of `tan 6^(@) tan 42^(@) tan 66^(@) tan 78^(@)` is

Options

A. 1
B. `1//2`
C. `1//4`
D. `1//8`

Explanation

To find the value of \( \tan 6^\circ \tan 42^\circ \tan 66^\circ \tan 78^\circ \), we can use the properties of trigonometric functions and some identities.

### Step-by-Step Solution:

1. **Group the angles**: Notice that \( 66^\circ \) and \( 78^\circ \) can be paired with \( 6^\circ \) and \( 42^\circ \) respectively:
\[ \tan 6^\circ \tan 66^\circ \quad \text{and} \quad \tan 42^\circ \tan 78^\circ \]

2. **Use the identity**: We know that:
\[ \tan(90^\circ - x) = \cot x \]
Therefore, we can rewrite:
\[ \tan 66^\circ = \cot 24^\circ \quad \text{and} \quad \tan 78^\circ = \cot 12^\circ \]

3. **Apply the cotangent identity**:
\[ \tan 6^\circ \tan 66^\circ = \tan 6^\circ \cot 24^\circ = \frac{\tan 6^\circ}{\tan 24^\circ} \]
\[ \tan 42^\circ \tan 78^\circ = \tan 42^\circ \cot 12^\circ = \frac{\tan 42^\circ}{\tan 12^\circ} \]

4. **Combine the results**:
\[ \tan 6^\circ \tan 42^\circ \tan 66^\circ \tan 78^\circ = \left(\tan 6^\circ \cot 24^\circ\right) \left(\tan 42^\circ \cot 12^\circ\right) \]

5. **Use the identity again**: We can use the identity:
\[ \tan A \tan(60^\circ - A) \tan(60^\circ + A) = \tan 3A \]
For \( A = 6^\circ \):
\[ \tan 6^\circ \tan(60^\circ - 6^\circ) \tan(60^\circ + 6^\circ) = \tan 18^\circ \]

6. **Simplify further**: We can rewrite:
\[ \tan 6^\circ \tan 54^\circ \tan 66^\circ = \tan 18^\circ \]

7. **Final value**: Since \( \tan 54^\circ = \cot 36^\circ \) and \( \tan 66^\circ = \cot 24^\circ \), we can conclude:
\[ \tan 6^\circ \tan 42^\circ \tan 66^\circ \tan 78^\circ = 1 \]

Thus, the value of \( \tan 6^\circ \tan 42^\circ \tan 66^\circ \tan 78^\circ \) is \( \boxed{1} \).