The value of `cos ( pi //5) cos ( 2pi //5) cos ( 4pi //5) cos ( 8pi //5)` is

Options

A. `1//16`
B. 0
C. `-1//8`
D. `-1//16`

Answer

To find the value of \( \cos \left( \frac{\pi}{5} \right) \cos \left( \frac{2\pi}{5} \right) \cos \left( \frac{4\pi}{5} \right) \cos \left( \frac{8\pi}{5} \right) \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \cos \left( \frac{\pi}{5} \right) \cos \left( \frac{2\pi}{5} \right) \cos \left( \frac{4\pi}{5} \right) \cos \left( \frac{8\pi}{5} \right) \]

### Step 2: Use the identity
We can use the identity \( 2 \sin x \cos x = \sin(2x) \). We will multiply and divide by \( 2 \sin \left( \frac{\pi}{5} \right) \):
\[ = \frac{2 \sin \left( \frac{\pi}{5} \right) \cos \left( \frac{\pi}{5} \right) \cos \left( \frac{2\pi}{5} \right) \cos \left( \frac{4\pi}{5} \right) \cos \left( \frac{8\pi}{5} \right)}{2 \sin \left( \frac{\pi}{5} \right)} \]
This gives us:
\[ = \frac{\sin \left( \frac{2\pi}{5} \right) \cos \left( \frac{2\pi}{5} \right) \cos \left( \frac{4\pi}{5} \right) \cos \left( \frac{8\pi}{5} \right)}{2 \sin \left( \frac{\pi}{5} \right)} \]

### Step 3: Simplify further
Now, we can apply the same identity again to \( \sin \left( \frac{2\pi}{5} \right) \):
\[ = \frac{\sin \left( \frac{4\pi}{5} \right) \cos \left( \frac{4\pi}{5} \right) \cos \left( \frac{8\pi}{5} \right)}{4 \sin \left( \frac{\pi}{5} \right)} \]

### Step 4: Continue simplifying
Next, we apply the identity to \( \sin \left( \frac{4\pi}{5} \right) \):
\[ = \frac{\sin \left( \frac{8\pi}{5} \right) \cos \left( \frac{8\pi}{5} \right)}{8 \sin \left( \frac{\pi}{5} \right)} \]

### Step 5: Final simplification
Now, we know that \( \sin \left( \frac{8\pi}{5} \right) = \sin \left( 3\pi + \frac{\pi}{5} \right) = -\sin \left( \frac{\pi}{5} \right) \):
\[ = \frac{-\sin \left( \frac{\pi}{5} \right) \cos \left( \frac{8\pi}{5} \right)}{8 \sin \left( \frac{\pi}{5} \right)} \]
The \( \sin \left( \frac{\pi}{5} \right) \) cancels out:
\[ = \frac{-\cos \left( \frac{8\pi}{5} \right)}{8} \]

### Step 6: Evaluate \( \cos \left( \frac{8\pi}{5} \right) \)
Since \( \cos \left( \frac{8\pi}{5} \right) = \cos \left( 3\pi + \frac{3\pi}{5} \right) = -\cos \left( \frac{3\pi}{5} \right) \):
\[ = \frac{-(-\cos \left( \frac{2\pi}{5} \right))}{8} \]
Thus, we have:
\[ = \frac{1}{16} \]

### Final Answer
The value of \( \cos \left( \frac{\pi}{5} \right) \cos \left( \frac{2\pi}{5} \right) \cos \left( \frac{4\pi}{5} \right) \cos \left( \frac{8\pi}{5} \right) \) is:
\[ \boxed{-\frac{1}{16}} \]