Solve
`cos(pi/9)+cos(pi/3)+cos((5pi)/(9))+cos((7pi)/(9))`?

Options

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

Answer

To solve the expression \( \cos\left(\frac{\pi}{9}\right) + \cos\left(\frac{\pi}{3}\right) + \cos\left(\frac{5\pi}{9}\right) + \cos\left(\frac{7\pi}{9}\right) \), we will follow these steps:

### Step 1: Calculate \( \cos\left(\frac{\pi}{3}\right) \)
We know that:
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

### Step 2: Substitute \( \cos\left(\frac{\pi}{3}\right) \) into the expression
Now, substituting this value into the expression gives us:
\[ \cos\left(\frac{\pi}{9}\right) + \frac{1}{2} + \cos\left(\frac{5\pi}{9}\right) + \cos\left(\frac{7\pi}{9}\right) \]

### Step 3: Use the cosine addition formula
Next, we will use the cosine addition formula:
\[ \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
We will apply this to \( \cos\left(\frac{5\pi}{9}\right) + \cos\left(\frac{7\pi}{9}\right) \).

Let \( A = \frac{5\pi}{9} \) and \( B = \frac{7\pi}{9} \):
\[ \cos\left(\frac{5\pi}{9}\right) + \cos\left(\frac{7\pi}{9}\right) = 2 \cos\left(\frac{\frac{5\pi}{9} + \frac{7\pi}{9}}{2}\right) \cos\left(\frac{\frac{5\pi}{9} - \frac{7\pi}{9}}{2}\right) \]

Calculating the averages:
\[ \frac{5\pi/9 + 7\pi/9}{2} = \frac{12\pi/9}{2} = \frac{2\pi}{3} \]
\[ \frac{5\pi/9 - 7\pi/9}{2} = \frac{-2\pi/9}{2} = -\frac{\pi}{9} \]

Thus,
\[ \cos\left(\frac{5\pi}{9}\right) + \cos\left(\frac{7\pi}{9}\right) = 2 \cos\left(\frac{2\pi}{3}\right) \cos\left(-\frac{\pi}{9}\right) \]

### Step 4: Calculate \( \cos\left(\frac{2\pi}{3}\right) \)
We know that:
\[ \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \]

### Step 5: Substitute back into the expression
Now substituting back, we have:
\[ \cos\left(\frac{5\pi}{9}\right) + \cos\left(\frac{7\pi}{9}\right) = 2 \left(-\frac{1}{2}\right) \cos\left(\frac{\pi}{9}\right) = -\cos\left(\frac{\pi}{9}\right) \]

### Step 6: Combine all terms
Now substituting this back into our expression:
\[ \cos\left(\frac{\pi}{9}\right) + \frac{1}{2} - \cos\left(\frac{\pi}{9}\right) \]
This simplifies to:
\[ \frac{1}{2} \]

### Final Answer
Thus, the final answer is:
\[ \cos\left(\frac{\pi}{9}\right) + \cos\left(\frac{\pi}{3}\right) + \cos\left(\frac{5\pi}{9}\right) + \cos\left(\frac{7\pi}{9}\right) = \frac{1}{2} \]

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