If `alpha=(pi)/(8)`, what is the value of `cosalphacos2alphacos4alpha`?

Options

A. 0
B. `(1)/(4)`
C. 8
D. 4

Explanation

To find the value of \( \cos \alpha \cos 2\alpha \cos 4\alpha \) when \( \alpha = \frac{\pi}{8} \), we can follow these steps:

### Step 1: Substitute \( \alpha \)
Given \( \alpha = \frac{\pi}{8} \), we substitute this value into the expression:
\[ \cos \alpha \cos 2\alpha \cos 4\alpha = \cos \left(\frac{\pi}{8}\right) \cos \left(2 \cdot \frac{\pi}{8}\right) \cos \left(4 \cdot \frac{\pi}{8}\right) \]

### Step 2: Simplify the angles
Now, we simplify the angles:
\[ \cos \left(2 \cdot \frac{\pi}{8}\right) = \cos \left(\frac{\pi}{4}\right) \quad \text{and} \quad \cos \left(4 \cdot \frac{\pi}{8}\right) = \cos \left(\frac{\pi}{2}\right) \]
Thus, we can rewrite the expression as:
\[ \cos \left(\frac{\pi}{8}\right) \cos \left(\frac{\pi}{4}\right) \cos \left(\frac{\pi}{2}\right) \]

### Step 3: Evaluate \( \cos \left(\frac{\pi}{2}\right) \)
We know that:
\[ \cos \left(\frac{\pi}{2}\right) = 0 \]

### Step 4: Substitute back into the expression
Substituting this value back into our expression gives:
\[ \cos \left(\frac{\pi}{8}\right) \cos \left(\frac{\pi}{4}\right) \cdot 0 = 0 \]

### Conclusion
Therefore, the value of \( \cos \alpha \cos 2\alpha \cos 4\alpha \) is:
\[ \boxed{0} \]