Evaluate : `tan 12^@ tan 38^@ tan 52^@ tan 60^@ tan 78^@`

Options

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

Answer

To evaluate the expression \( \tan 12^\circ \tan 38^\circ \tan 52^\circ \tan 60^\circ \tan 78^\circ \), we can use the properties of tangent and cotangent.

### Step-by-Step Solution:

1. **Identify Complementary Angles:**
We know that \( \tan(90^\circ - \theta) = \cot(\theta) \). Therefore, we can rewrite some of the angles:
- \( \tan 52^\circ = \tan(90^\circ - 38^\circ) = \cot 38^\circ \)
- \( \tan 78^\circ = \tan(90^\circ - 12^\circ) = \cot 12^\circ \)

2. **Rewrite the Expression:**
Substitute the complementary angles into the original expression:
\[ \tan 12^\circ \tan 38^\circ \tan 52^\circ \tan 60^\circ \tan 78^\circ = \tan 12^\circ \tan 38^\circ \cot 38^\circ \tan 60^\circ \cot 12^\circ \]

3. **Use the Identity \( \tan \theta \cdot \cot \theta = 1 \):**
We can now group the terms:
\[ (\tan 12^\circ \cot 12^\circ) \cdot (\tan 38^\circ \cot 38^\circ) \cdot \tan 60^\circ \]
Each of the products \( \tan \theta \cdot \cot \theta = 1 \):
\[ 1 \cdot 1 \cdot \tan 60^\circ \]

4. **Evaluate \( \tan 60^\circ \):**
We know that \( \tan 60^\circ = \sqrt{3} \).

5. **Final Calculation:**
Therefore, the entire expression simplifies to:
\[ 1 \cdot 1 \cdot \sqrt{3} = \sqrt{3} \]

### Final Answer:
\[ \tan 12^\circ \tan 38^\circ \tan 52^\circ \tan 60^\circ \tan 78^\circ = \sqrt{3} \]