What is the value of `cos(pi//9)+cos(pi//3)+cos(5pi//9)+cos(7pi//9)`?

Options

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

Explanation

To find the value of \( \cos\left(\frac{\pi}{9}\right) + \cos\left(\frac{\pi}{3}\right) + \cos\left(\frac{5\pi}{9}\right) + \cos\left(\frac{7\pi}{9}\right) \), we can follow these steps:

### Step 1: Substitute the known value
We know that \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \). So, we can rewrite the expression as:
\[ \cos\left(\frac{\pi}{9}\right) + \frac{1}{2} + \cos\left(\frac{5\pi}{9}\right) + \cos\left(\frac{7\pi}{9}\right) \]

### Step 2: Group the cosines
Now, we can group \( \cos\left(\frac{\pi}{9}\right) \) with \( \cos\left(\frac{5\pi}{9}\right) \) and \( \cos\left(\frac{7\pi}{9}\right) \):
\[ \frac{1}{2} + \left( \cos\left(\frac{\pi}{9}\right) + \cos\left(\frac{5\pi}{9}\right) + \cos\left(\frac{7\pi}{9}\right) \right) \]

### Step 3: Apply the cosine addition formula
Using the formula \( \cos c + \cos d = 2 \cos\left(\frac{c+d}{2}\right) \cos\left(\frac{c-d}{2}\right) \), we can combine \( \cos\left(\frac{\pi}{9}\right) \) and \( \cos\left(\frac{7\pi}{9}\right) \):
\[ \cos\left(\frac{\pi}{9}\right) + \cos\left(\frac{7\pi}{9}\right) = 2 \cos\left(\frac{\frac{\pi}{9} + \frac{7\pi}{9}}{2}\right) \cos\left(\frac{\frac{\pi}{9} - \frac{7\pi}{9}}{2}\right) \]
Calculating the averages:
\[ \frac{\frac{\pi}{9} + \frac{7\pi}{9}}{2} = \frac{4\pi}{9} \quad \text{and} \quad \frac{\frac{\pi}{9} - \frac{7\pi}{9}}{2} = -\frac{3\pi}{9} = -\frac{\pi}{3} \]
Thus,
\[ \cos\left(\frac{\pi}{9}\right) + \cos\left(\frac{7\pi}{9}\right) = 2 \cos\left(\frac{4\pi}{9}\right) \cos\left(-\frac{\pi}{3}\right) \]
Since \( \cos(-\theta) = \cos(\theta) \), we have:
\[ \cos\left(-\frac{\pi}{3}\right) = \frac{1}{2} \]
So,
\[ \cos\left(\frac{\pi}{9}\right) + \cos\left(\frac{7\pi}{9}\right) = 2 \cos\left(\frac{4\pi}{9}\right) \cdot \frac{1}{2} = \cos\left(\frac{4\pi}{9}\right) \]

### Step 4: Combine with the remaining cosine
Now we can substitute this back into our expression:
\[ \frac{1}{2} + \cos\left(\frac{4\pi}{9}\right) + \cos\left(\frac{5\pi}{9}\right) \]
Next, we can combine \( \cos\left(\frac{4\pi}{9}\right) \) and \( \cos\left(\frac{5\pi}{9}\right) \):
\[ \cos\left(\frac{4\pi}{9}\right) + \cos\left(\frac{5\pi}{9}\right) = 2 \cos\left(\frac{\frac{4\pi}{9} + \frac{5\pi}{9}}{2}\right) \cos\left(\frac{\frac{4\pi}{9} - \frac{5\pi}{9}}{2}\right) \]
Calculating the averages:
\[ \frac{\frac{4\pi}{9} + \frac{5\pi}{9}}{2} = \frac{9\pi}{18} = \frac{\pi}{2} \quad \text{and} \quad \frac{\frac{4\pi}{9} - \frac{5\pi}{9}}{2} = -\frac{\pi}{18} \]
Thus,
\[ \cos\left(\frac{4\pi}{9}\right) + \cos\left(\frac{5\pi}{9}\right) = 2 \cos\left(\frac{\pi}{2}\right) \cos\left(-\frac{\pi}{18}\right) \]
Since \( \cos\left(\frac{\pi}{2}\right) = 0 \), we find:
\[ \cos\left(\frac{4\pi}{9}\right) + \cos\left(\frac{5\pi}{9}\right) = 0 \]

### Step 5: Final result
Thus, we have:
\[ \frac{1}{2} + 0 = \frac{1}{2} \]

### Final Answer
The value of \( \cos\left(\frac{\pi}{9}\right) + \cos\left(\frac{\pi}{3}\right) + \cos\left(\frac{5\pi}{9}\right) + \cos\left(\frac{7\pi}{9}\right) \) is \( \frac{1}{2} \).