Evaluate : ` tan ""(pi )/(12) .tan ""( pi )/(16) .tan ""(5pi )/(12) .tan ""( 7pi )/(16)`

Options

A. `-1`
B. ` 1`
C. `0`
D. None of these

Answer

To evaluate the expression \( \tan \left( \frac{\pi}{12} \right) \cdot \tan \left( \frac{\pi}{16} \right) \cdot \tan \left( \frac{5\pi}{12} \right) \cdot \tan \left( \frac{7\pi}{16} \right) \), we can use the properties of the tangent function and the cotangent function.

### Step-by-step Solution:

1. **Rewrite the Tangent Functions**:
We can use the identity \( \tan\left(\frac{\pi}{2} - \theta\right) = \cot(\theta) \) to rewrite some of the tangent functions:
- \( \tan\left(\frac{5\pi}{12}\right) = \tan\left(\frac{\pi}{2} - \frac{7\pi}{12}\right) = \cot\left(\frac{7\pi}{12}\right) \)
- \( \tan\left(\frac{7\pi}{16}\right) = \tan\left(\frac{\pi}{2} - \frac{7\pi}{16}\right) = \cot\left(\frac{7\pi}{16}\right) \)

2. **Apply the Cotangent Identity**:
Using the identity \( \tan(\theta) \cdot \cot(\theta) = 1 \):
- \( \tan\left(\frac{\pi}{12}\right) \cdot \cot\left(\frac{5\pi}{12}\right) = 1 \)
- \( \tan\left(\frac{\pi}{16}\right) \cdot \cot\left(\frac{7\pi}{16}\right) = 1 \)

3. **Combine the Results**:
Therefore, we can combine the results:
\[ \tan\left(\frac{\pi}{12}\right) \cdot \tan\left(\frac{\pi}{16}\right) \cdot \tan\left(\frac{5\pi}{12}\right) \cdot \tan\left(\frac{7\pi}{16}\right) = 1 \cdot 1 = 1 \]

4. **Final Result**:
Thus, the value of the expression is:
\[ \boxed{1} \]