The value of `cos""(pi)/(9)cos""(2pi)/(9)cos""(3pi)/(9)cos""(4pi)/(9),` is

Options

A. `1/8`
B. `1/16`
C. `1/64`
D. `1/4`

Answer

To find the value of \( \cos\left(\frac{\pi}{9}\right) \cos\left(\frac{2\pi}{9}\right) \cos\left(\frac{3\pi}{9}\right) \cos\left(\frac{4\pi}{9}\right) \), we can follow these steps:

### Step 1: Simplify \( \cos\left(\frac{3\pi}{9}\right) \)
First, we simplify \( \cos\left(\frac{3\pi}{9}\right) \):
\[ \cos\left(\frac{3\pi}{9}\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

### Step 2: Rewrite the expression
Now, we can rewrite the expression:
\[ \cos\left(\frac{\pi}{9}\right) \cos\left(\frac{2\pi}{9}\right) \cos\left(\frac{3\pi}{9}\right) \cos\left(\frac{4\pi}{9}\right) = \cos\left(\frac{\pi}{9}\right) \cos\left(\frac{2\pi}{9}\right) \cdot \frac{1}{2} \cdot \cos\left(\frac{4\pi}{9}\right) \]
This simplifies to:
\[ \frac{1}{2} \cos\left(\frac{\pi}{9}\right) \cos\left(\frac{2\pi}{9}\right) \cos\left(\frac{4\pi}{9}\right) \]

### Step 3: Use the product-to-sum identities
Next, we can use the product-to-sum identities to simplify \( \cos\left(\frac{\pi}{9}\right) \cos\left(\frac{4\pi}{9}\right) \):
\[ \cos A \cos B = \frac{1}{2} \left( \cos(A+B) + \cos(A-B) \right) \]
Here, let \( A = \frac{\pi}{9} \) and \( B = \frac{4\pi}{9} \):
\[ \cos\left(\frac{\pi}{9}\right) \cos\left(\frac{4\pi}{9}\right) = \frac{1}{2} \left( \cos\left(\frac{5\pi}{9}\right) + \cos\left(-\frac{3\pi}{9}\right) \right) \]
Since \( \cos(-x) = \cos(x) \):
\[ \cos\left(-\frac{3\pi}{9}\right) = \cos\left(\frac{3\pi}{9}\right) = \frac{1}{2} \]
So, we have:
\[ \cos\left(\frac{\pi}{9}\right) \cos\left(\frac{4\pi}{9}\right) = \frac{1}{2} \left( \cos\left(\frac{5\pi}{9}\right) + \frac{1}{2} \right) \]

### Step 4: Substitute back into the expression
Now substituting back:
\[ \frac{1}{2} \cdot \frac{1}{2} \left( \cos\left(\frac{5\pi}{9}\right) + \frac{1}{2} \right) \cos\left(\frac{2\pi}{9}\right) \]
This becomes:
\[ \frac{1}{4} \left( \cos\left(\frac{5\pi}{9}\right) + \frac{1}{2} \right) \cos\left(\frac{2\pi}{9}\right) \]

### Step 5: Evaluate \( \cos\left(\frac{5\pi}{9}\right) \)
Using the identity \( \cos\left(\frac{5\pi}{9}\right) = -\cos\left(\frac{4\pi}{9}\right) \):
\[ \cos\left(\frac{5\pi}{9}\right) + \frac{1}{2} = -\cos\left(\frac{4\pi}{9}\right) + \frac{1}{2} \]
This results in:
\[ \frac{1}{4} \left(-\cos\left(\frac{4\pi}{9}\right) + \frac{1}{2}\right) \cos\left(\frac{2\pi}{9}\right) \]

### Step 6: Final Calculation
After evaluating the above expression, we find:
\[ \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8} \]

Thus, the value of \( \cos\left(\frac{\pi}{9}\right) \cos\left(\frac{2\pi}{9}\right) \cos\left(\frac{3\pi}{9}\right) \cos\left(\frac{4\pi}{9}\right) \) is:
\[ \frac{1}{16} \]

### Final Answer
The value is \( \frac{1}{16} \).