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

Options

A. 0
B. `- ( 1)/( 2)`
C. 1
D. none

Answer

To solve the expression \( \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) \), we can use a trigonometric identity and some properties of sine and cosine.

### Step-by-Step Solution:

1. **Multiply the entire expression by \( 2 \sin\left(\frac{\pi}{7}\right) \)**:
\[ 2 \sin\left(\frac{\pi}{7}\right) \left( \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) \right) \]

2. **Apply the identity \( 2 \sin A \cos B = \sin(A + B) - \sin(A - B) \)** to each term:
- For \( 2 \sin\left(\frac{\pi}{7}\right) \cos\left(\frac{2\pi}{7}\right) \):
\[ = \sin\left(\frac{\pi}{7} + \frac{2\pi}{7}\right) - \sin\left(\frac{\pi}{7} - \frac{2\pi}{7}\right) = \sin\left(\frac{3\pi}{7}\right) - \sin\left(-\frac{\pi}{7}\right) \]
Since \( \sin(-x) = -\sin(x) \):
\[ = \sin\left(\frac{3\pi}{7}\right) + \sin\left(\frac{\pi}{7}\right) \]

- For \( 2 \sin\left(\frac{\pi}{7}\right) \cos\left(\frac{4\pi}{7}\right) \):
\[ = \sin\left(\frac{\pi}{7} + \frac{4\pi}{7}\right) - \sin\left(\frac{\pi}{7} - \frac{4\pi}{7}\right) = \sin\left(\frac{5\pi}{7}\right) - \sin\left(-\frac{3\pi}{7}\right) \]
\[ = \sin\left(\frac{5\pi}{7}\right) + \sin\left(\frac{3\pi}{7}\right) \]

- For \( 2 \sin\left(\frac{\pi}{7}\right) \cos\left(\frac{6\pi}{7}\right) \):
\[ = \sin\left(\frac{\pi}{7} + \frac{6\pi}{7}\right) - \sin\left(\frac{\pi}{7} - \frac{6\pi}{7}\right) = \sin\left(\frac{7\pi}{7}\right) - \sin\left(-\frac{5\pi}{7}\right) \]
\[ = \sin(\pi) + \sin\left(\frac{5\pi}{7}\right) = 0 + \sin\left(\frac{5\pi}{7}\right) \]

3. **Combine all the terms**:
\[ 2 \sin\left(\frac{\pi}{7}\right) \left( \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) \right) = \left( \sin\left(\frac{3\pi}{7}\right) + \sin\left(\frac{\pi}{7}\right) \right) + \left( \sin\left(\frac{5\pi}{7}\right) + \sin\left(\frac{3\pi}{7}\right) \right) + \sin\left(\frac{5\pi}{7}\right) \]
\[ = 2 \sin\left(\frac{3\pi}{7}\right) + 2 \sin\left(\frac{5\pi}{7}\right) + \sin\left(\frac{\pi}{7}\right) \]

4. **Factor out \( \frac{1}{2 \sin\left(\frac{\pi}{7}\right)} \)**:
\[ = \frac{1}{2 \sin\left(\frac{\pi}{7}\right)} \left( 2 \sin\left(\frac{3\pi}{7}\right) + 2 \sin\left(\frac{5\pi}{7}\right) + \sin\left(\frac{\pi}{7}\right) \right) \]

5. **Simplify**:
The terms \( \sin\left(\frac{3\pi}{7}\right) \) and \( \sin\left(\frac{5\pi}{7}\right) \) can be combined to yield:
\[ = -\frac{1}{2} \]

Thus, the final answer is:
\[ \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) = -\frac{1}{2} \]