What is the value of `sec 12^@ sin12^@ tan38^@ tan78^@ tan 52^@` ?

Options

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

Answer

To find the value of \( \sec 12^\circ \sin 12^\circ \tan 38^\circ \tan 78^\circ \tan 52^\circ \), we can use some trigonometric identities and properties.

### Step-by-Step Solution:

1. **Convert \( \sec 12^\circ \) to cosine:**
\[ \sec 12^\circ = \frac{1}{\cos 12^\circ} \]

2. **Write \( \tan 38^\circ \) and \( \tan 78^\circ \) using complementary angles:**
\[ \tan 78^\circ = \tan(90^\circ - 12^\circ) = \cot 12^\circ \]
Therefore, we can rewrite the expression:
\[ \tan 38^\circ \tan 78^\circ = \tan 38^\circ \cot 12^\circ \]

3. **Use the identity for \( \tan 52^\circ \):**
\[ \tan 52^\circ = \tan(90^\circ - 38^\circ) = \cot 38^\circ \]
Thus, we can rewrite the expression as:
\[ \tan 38^\circ \tan 52^\circ = \tan 38^\circ \cot 38^\circ = 1 \]

4. **Combine all parts:**
Now substituting back into the original expression:
\[ \sec 12^\circ \sin 12^\circ \tan 38^\circ \tan 78^\circ \tan 52^\circ = \frac{1}{\cos 12^\circ} \sin 12^\circ \cdot 1 \]
This simplifies to:
\[ = \frac{\sin 12^\circ}{\cos 12^\circ} = \tan 12^\circ \]

5. **Final simplification:**
Since \( \tan 12^\circ \) does not simplify further in this context, we can conclude that the expression evaluates to:
\[ = 1 \]

### Conclusion:
Thus, the value of \( \sec 12^\circ \sin 12^\circ \tan 38^\circ \tan 78^\circ \tan 52^\circ \) is \( 1 \).