The value of `cos"" ( pi )/( 7) cos"" ( 2pi )/(7) cos "" ( 4pi )/( 7 )` is

Options

A. 0
B. `1//2`
C. `1//3`
D. `- 1//8`

Answer

To find the value of \( \cos\left(\frac{\pi}{7}\right) \cos\left(\frac{2\pi}{7}\right) \cos\left(\frac{4\pi}{7}\right) \), we can follow these steps:

### Step 1: Multiply and Divide by \( 2 \sin\left(\frac{\pi}{7}\right) \)
We start with the expression:
\[ \cos\left(\frac{\pi}{7}\right) \cos\left(\frac{2\pi}{7}\right) \cos\left(\frac{4\pi}{7}\right) \]
We multiply and divide by \( 2 \sin\left(\frac{\pi}{7}\right) \):
\[ \frac{2 \sin\left(\frac{\pi}{7}\right) \cos\left(\frac{\pi}{7}\right) \cos\left(\frac{2\pi}{7}\right) \cos\left(\frac{4\pi}{7}\right)}{2 \sin\left(\frac{\pi}{7}\right)} \]

### Step 2: Use the Identity \( 2 \sin \theta \cos \theta = \sin(2\theta) \)
The numerator can be simplified using the identity \( 2 \sin \theta \cos \theta = \sin(2\theta) \):
\[ \frac{\sin\left(\frac{2\pi}{7}\right) \cos\left(\frac{2\pi}{7}\right) \cos\left(\frac{4\pi}{7}\right)}{2 \sin\left(\frac{\pi}{7}\right)} \]

### Step 3: Apply the Identity Again
Now, we can apply the identity again to \( \sin\left(\frac{2\pi}{7}\right) \cos\left(\frac{2\pi}{7}\right) \):
\[ \frac{\sin\left(\frac{4\pi}{7}\right) \cos\left(\frac{4\pi}{7}\right)}{4 \sin\left(\frac{\pi}{7}\right)} \]

### Step 4: Use the Identity Once More
We apply the identity again:
\[ \frac{\sin\left(\frac{8\pi}{7}\right)}{8 \sin\left(\frac{\pi}{7}\right)} \]

### Step 5: Simplify \( \sin\left(\frac{8\pi}{7}\right) \)
Now, we simplify \( \sin\left(\frac{8\pi}{7}\right) \):
\[ \sin\left(\frac{8\pi}{7}\right) = \sin\left(\pi + \frac{\pi}{7}\right) = -\sin\left(\frac{\pi}{7}\right) \]

### Step 6: Substitute Back
Substituting back into our expression gives:
\[ \frac{-\sin\left(\frac{\pi}{7}\right)}{8 \sin\left(\frac{\pi}{7}\right)} \]

### Step 7: Cancel \( \sin\left(\frac{\pi}{7}\right) \)
Assuming \( \sin\left(\frac{\pi}{7}\right) \neq 0 \), we can cancel \( \sin\left(\frac{\pi}{7}\right) \):
\[ -\frac{1}{8} \]

### Final Answer
Thus, the value of \( \cos\left(\frac{\pi}{7}\right) \cos\left(\frac{2\pi}{7}\right) \cos\left(\frac{4\pi}{7}\right) \) is:
\[ \boxed{-\frac{1}{8}} \]