What of the value of `tan.(pi)/(8).tan.(pi)/(12).tan.(pi)/(4).tan.(3pi)/(8)tan.(5pi)/(12)` ?

Options

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

Answer

To find the value of the expression \( \tan\left(\frac{\pi}{8}\right) \tan\left(\frac{\pi}{12}\right) \tan\left(\frac{\pi}{4}\right) \tan\left(\frac{3\pi}{8}\right) \tan\left(\frac{5\pi}{12}\right) \), we can use the property of tangent that states:

\[ \tan(x) \tan\left(\frac{\pi}{2} - x\right) = 1 \]

### Step-by-Step Solution:

1. **Identify pairs of angles that sum to \( \frac{\pi}{2} \)**:
- The first pair is \( \tan\left(\frac{\pi}{8}\right) \) and \( \tan\left(\frac{3\pi}{8}\right) \).
- Check their sum:
\[ \frac{\pi}{8} + \frac{3\pi}{8} = \frac{4\pi}{8} = \frac{\pi}{2} \]
- Therefore, \( \tan\left(\frac{\pi}{8}\right) \tan\left(\frac{3\pi}{8}\right) = 1 \).

2. **Identify the second pair**:
- The second pair is \( \tan\left(\frac{\pi}{12}\right) \) and \( \tan\left(\frac{5\pi}{12}\right) \).
- Check their sum:
\[ \frac{\pi}{12} + \frac{5\pi}{12} = \frac{6\pi}{12} = \frac{\pi}{2} \]
- Therefore, \( \tan\left(\frac{\pi}{12}\right) \tan\left(\frac{5\pi}{12}\right) = 1 \).

3. **Evaluate \( \tan\left(\frac{\pi}{4}\right) \)**:
- We know that:
\[ \tan\left(\frac{\pi}{4}\right) = 1 \]

4. **Combine all parts**:
- Now we can combine all the results:
\[ \tan\left(\frac{\pi}{8}\right) \tan\left(\frac{3\pi}{8}\right) \tan\left(\frac{\pi}{12}\right) \tan\left(\frac{5\pi}{12}\right) \tan\left(\frac{\pi}{4}\right) = 1 \cdot 1 \cdot 1 = 1 \]

### Final Answer:
The value of \( \tan\left(\frac{\pi}{8}\right) \tan\left(\frac{\pi}{12}\right) \tan\left(\frac{\pi}{4}\right) \tan\left(\frac{3\pi}{8}\right) \tan\left(\frac{5\pi}{12}\right) \) is **1**.

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