`cos""(pi)/(7)+cos""(2pi)/(7)+cos""(3pi)/(7)+cos""(4pi)/(7)+cos""(5pi)/(7)+cos""(6pi)/(7)`

Options

A. 0
B. 1
C. `-1`
D. 2

Answer

To solve the problem, we need to find the value of the expression:

\[ S = \cos\left(\frac{\pi}{7}\right) + \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{3\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{5\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) \]

### Step 1: Use Symmetry of Cosine
Notice that \(\cos\left(\frac{6\pi}{7}\right) = -\cos\left(\frac{\pi}{7}\right)\), \(\cos\left(\frac{5\pi}{7}\right) = -\cos\left(\frac{2\pi}{7}\right)\), and \(\cos\left(\frac{4\pi}{7}\right) = -\cos\left(\frac{3\pi}{7}\right)\). This means we can pair the terms in the sum:

\[ S = \left(\cos\left(\frac{\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right)\right) + \left(\cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{5\pi}{7}\right)\right) + \left(\cos\left(\frac{3\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right)\right) \]

### Step 2: Simplify Each Pair
Using the properties of cosine, we can simplify each pair:

1. \(\cos\left(\frac{\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) = 0\)
2. \(\cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{5\pi}{7}\right) = 0\)
3. \(\cos\left(\frac{3\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) = 0\)

### Step 3: Combine the Results
Thus, we have:

\[ S = 0 + 0 + 0 = 0 \]

### Final Result
Therefore, the value of the expression is:

\[ \boxed{0} \]