Find the value of `cos((2pi)/7)+cos((4pi)/7)+cos((6pi)/7)`

Answer

To find the value of \( \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) \), we can follow these steps:

### Step 1: Multiply and Divide by \( 2\sin\left(\frac{\pi}{7}\right) \)
We start by multiplying and dividing the expression by \( 2\sin\left(\frac{\pi}{7}\right) \):

\[ \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) = \frac{1}{2\sin\left(\frac{\pi}{7}\right)} \cdot \left( 2\sin\left(\frac{\pi}{7}\right) \cos\left(\frac{2\pi}{7}\right) + 2\sin\left(\frac{\pi}{7}\right) \cos\left(\frac{4\pi}{7}\right) + 2\sin\left(\frac{\pi}{7}\right) \cos\left(\frac{6\pi}{7}\right) \right) \]

### Step 2: Use the Formula for \( 2\sin A \cos B \)
Using the identity \( 2\sin A \cos B = \sin(A+B) + \sin(A-B) \):

\[ = \frac{1}{2\sin\left(\frac{\pi}{7}\right)} \left( \sin\left(\frac{3\pi}{7}\right) + \sin\left(-\frac{\pi}{7}\right) + \sin\left(\frac{5\pi}{7}\right) + \sin\left(-\frac{3\pi}{7}\right) + \sin\left(\pi\right) + \sin\left(-\frac{5\pi}{7}\right) \right) \]

### Step 3: Simplify Using the Identity \( \sin(-\theta) = -\sin(\theta) \)
Applying the identity \( \sin(-\theta) = -\sin(\theta) \):

\[ = \frac{1}{2\sin\left(\frac{\pi}{7}\right)} \left( \sin\left(\frac{3\pi}{7}\right) - \sin\left(\frac{\pi}{7}\right) + \sin\left(\frac{5\pi}{7}\right) - \sin\left(\frac{3\pi}{7}\right) + 0 - \sin\left(\frac{5\pi}{7}\right) \right) \]

### Step 4: Cancel Out Terms
Notice that \( \sin\left(\frac{3\pi}{7}\right) \) and \( -\sin\left(\frac{3\pi}{7}\right) \) cancel out, as do \( \sin\left(\frac{5\pi}{7}\right) \) and \( -\sin\left(\frac{5\pi}{7}\right) \):

\[ = \frac{1}{2\sin\left(\frac{\pi}{7}\right)} \left( -\sin\left(\frac{\pi}{7}\right) \right) \]

### Step 5: Final Simplification
This simplifies to:

\[ = -\frac{1}{2} \]

Thus, the value of \( \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) \) is:

\[ \boxed{-\frac{1}{2}} \]