`cos""(pi)/(15).cos""(2pi)/(15).cos""(4pi)/(15).cos""(8pi)/(15)` is equal to

Options

A. (a) `(1)/(16)`
B. (b) `-(1)/(16)`
C. (c) 1
D. (d) 0

Answer

To solve the expression \( \cos\left(\frac{\pi}{15}\right) \cdot \cos\left(\frac{2\pi}{15}\right) \cdot \cos\left(\frac{4\pi}{15}\right) \cdot \cos\left(\frac{8\pi}{15}\right) \), we can use a known formula for the product of cosines.

### Step-by-Step Solution:

1. **Identify the angles**: The angles in the product are \( \frac{\pi}{15}, \frac{2\pi}{15}, \frac{4\pi}{15}, \frac{8\pi}{15} \). We can denote \( A = \frac{\pi}{15} \).

2. **Recognize the pattern**: The angles can be expressed as \( A, 2A, 4A, 8A \). This follows the pattern of \( A, 2A, 2^2A, 2^3A \).

3. **Use the product-to-sum formula**: There is a known result for the product of cosines:
\[ \prod_{k=0}^{n-1} \cos(2^k A) = \frac{\sin(2^n A)}{2^n \sin(A)} \]
Here, \( n = 4 \) (since we have four terms) and \( A = \frac{\pi}{15} \).

4. **Apply the formula**: Substitute \( n = 4 \) and \( A = \frac{\pi}{15} \) into the formula:
\[ \cos\left(\frac{\pi}{15}\right) \cdot \cos\left(\frac{2\pi}{15}\right) \cdot \cos\left(\frac{4\pi}{15}\right) \cdot \cos\left(\frac{8\pi}{15}\right) = \frac{\sin(16 \cdot \frac{\pi}{15})}{2^4 \sin\left(\frac{\pi}{15}\right)} \]

5. **Simplify the sine term**: Calculate \( 16 \cdot \frac{\pi}{15} = \frac{16\pi}{15} \). We can express this as:
\[ \sin\left(\frac{16\pi}{15}\right) = \sin\left(\pi + \frac{\pi}{15}\right) = -\sin\left(\frac{\pi}{15}\right) \]
(using the identity \( \sin(\pi + x) = -\sin(x) \)).

6. **Substitute back into the equation**:
\[ \frac{-\sin\left(\frac{\pi}{15}\right)}{16 \sin\left(\frac{\pi}{15}\right)} = \frac{-1}{16} \]

7. **Final answer**: Therefore, the value of the original expression is:
\[ \cos\left(\frac{\pi}{15}\right) \cdot \cos\left(\frac{2\pi}{15}\right) \cdot \cos\left(\frac{4\pi}{15}\right) \cdot \cos\left(\frac{8\pi}{15}\right) = -\frac{1}{16} \]