The value of `cos""(pi)/(11)+cos""(3pi)/(11)+cos""(5pi)/(11)+cos""(7pi)/(11)+cos""(9pi)/(11),` is

Options

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

Answer

To find the value of the expression \( S = \cos\left(\frac{\pi}{11}\right) + \cos\left(\frac{3\pi}{11}\right) + \cos\left(\frac{5\pi}{11}\right) + \cos\left(\frac{7\pi}{11}\right) + \cos\left(\frac{9\pi}{11}\right) \), we can follow these steps:

### Step 1: Define the sum
Let
\[ S = \cos\left(\frac{\pi}{11}\right) + \cos\left(\frac{3\pi}{11}\right) + \cos\left(\frac{5\pi}{11}\right) + \cos\left(\frac{7\pi}{11}\right) + \cos\left(\frac{9\pi}{11}\right) \]

### Step 2: Use the identity for sum of cosines
We can use the formula for the sum of cosines:
\[ \sum_{k=0}^{n-1} \cos\left(a + kd\right) = \frac{\sin\left(\frac{nd}{2}\right) \cos\left(a + \frac{(n-1)d}{2}\right)}{\sin\left(\frac{d}{2}\right)} \]
In our case, \( a = \frac{\pi}{11} \), \( d = \frac{2\pi}{11} \), and \( n = 5 \).

### Step 3: Apply the formula
Substituting into the formula, we have:
\[ S = \frac{\sin\left(\frac{5 \cdot \frac{2\pi}{11}}{2}\right) \cos\left(\frac{\pi}{11} + \frac{4\pi}{11}\right)}{\sin\left(\frac{\frac{2\pi}{11}}{2}\right)} \]
This simplifies to:
\[ S = \frac{\sin\left(\frac{5\pi}{11}\right) \cos\left(\frac{5\pi}{11}\right)}{\sin\left(\frac{\pi}{11}\right)} \]

### Step 4: Use the double angle identity
Using the double angle identity for sine, we have:
\[ \sin(2x) = 2 \sin(x) \cos(x) \]
Thus, we can write:
\[ S = \frac{1}{2} \cdot \frac{\sin\left(\frac{10\pi}{11}\right)}{\sin\left(\frac{\pi}{11}\right)} \]

### Step 5: Simplify using sine properties
Since \( \sin\left(\frac{10\pi}{11}\right) = \sin\left(\pi - \frac{\pi}{11}\right) = \sin\left(\frac{\pi}{11}\right) \), we have:
\[ S = \frac{1}{2} \cdot \frac{\sin\left(\frac{\pi}{11}\right)}{\sin\left(\frac{\pi}{11}\right)} = \frac{1}{2} \]

### Final Answer
Thus, the value of \( S \) is:
\[ \boxed{\frac{1}{2}} \]