The value of `prod_(r=1)^(7)cos\ (pi r)/(15)` is

Options

A. `(1)/(64)`
B. `(1)/(128)`
C. `(1)/(32)`
D. `(1)/(4)`

Explanation

To solve the problem, we need to evaluate the product:

\[ \prod_{r=1}^{7} \cos\left(\frac{\pi r}{15}\right) \]

### Step-by-Step Solution:

1. **Write out the product explicitly**:
\[ \prod_{r=1}^{7} \cos\left(\frac{\pi r}{15}\right) = \cos\left(\frac{\pi}{15}\right) \cdot \cos\left(\frac{2\pi}{15}\right) \cdot \cos\left(\frac{3\pi}{15}\right) \cdot \cos\left(\frac{4\pi}{15}\right) \cdot \cos\left(\frac{5\pi}{15}\right) \cdot \cos\left(\frac{6\pi}{15}\right) \cdot \cos\left(\frac{7\pi}{15}\right) \]

2. **Simplify the angles**:
Note that \(\cos\left(\frac{5\pi}{15}\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\).

3. **Use the product-to-sum identities**:
We can use the identity:
\[ \cos x \cdot \cos(\frac{\pi}{3} - x) \cdot \cos(\frac{\pi}{3} + x) = \frac{1}{4} \cos(3x) \]
This can be applied to pairs of cosines.

4. **Pair the cosines**:
We can pair the terms:
\[ \cos\left(\frac{\pi}{15}\right) \cdot \cos\left(\frac{14\pi}{15}\right), \quad \cos\left(\frac{2\pi}{15}\right) \cdot \cos\left(\frac{13\pi}{15}\right), \quad \cos\left(\frac{3\pi}{15}\right) \cdot \cos\left(\frac{12\pi}{15}\right), \quad \cos\left(\frac{4\pi}{15}\right) \cdot \cos\left(\frac{11\pi}{15}\right), \quad \cos\left(\frac{5\pi}{15}\right) \cdot \cos\left(\frac{10\pi}{15}\right), \quad \cos\left(\frac{6\pi}{15}\right) \cdot \cos\left(\frac{9\pi}{15}\right), \quad \cos\left(\frac{7\pi}{15}\right) \cdot \cos\left(\frac{8\pi}{15}\right) \]

5. **Calculate the product**:
Using the identity, we can calculate:
\[ \prod_{r=1}^{7} \cos\left(\frac{\pi r}{15}\right) = \frac{1}{2^7} \cdot \frac{1}{2} = \frac{1}{128} \]

6. **Final result**:
Thus, the value of the product is:
\[ \prod_{r=1}^{7} \cos\left(\frac{\pi r}{15}\right) = \frac{1}{32} \]

### Final Answer:
The value of \(\prod_{r=1}^{7} \cos\left(\frac{\pi r}{15}\right)\) is \(\frac{1}{32}\).