Find the summation of the following
`(i)` `cos((2pi)/7) + cos((4pi)/7)+ cos((6pi)/7)`
`(ii)` `cos((pi)/7) + cos((2pi)/7)+ cos((3pi)/7)+ cos((4pi)/7)+ cos((5pi)/7)+ cos((6pi)/7)`
`(iii)` `cos (pi/11)+ cos((3pi)/11) + cos((5pi)/11)+ cos((7pi)/11) + cos((9pi)/11)`

Explanation

To solve the given problems, we will use properties of trigonometric functions and identities. Let's break it down step by step.

### (i) Find \( \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) \)

1. **Use the identity for the sum of cosines**:
\[ \cos x + \cos(2\pi - x) = 0 \]
Here, we can pair \( \cos\left(\frac{2\pi}{7}\right) \) with \( \cos\left(\frac{6\pi}{7}\right) \):
\[ \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) = 0 \]

2. **Evaluate the remaining term**:
The only term left is \( \cos\left(\frac{4\pi}{7}\right) \). Thus:
\[ \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) = 0 + \cos\left(\frac{4\pi}{7}\right) = \cos\left(\frac{4\pi}{7}\right) \]

3. **Final Result**:
\[ \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) = \cos\left(\frac{4\pi}{7}\right) \]

### (ii) Find \( \cos\left(\frac{\pi}{7}\right) + \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{3\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{5\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) \)

1. **Pair the cosines**:
Using the identity \( \cos x + \cos(2\pi - x) = 0 \):
\[ \cos\left(\frac{\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) = 0 \]
\[ \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{5\pi}{7}\right) = 0 \]
\[ \cos\left(\frac{3\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) = 0 \]

2. **Sum up the pairs**:
All pairs sum to zero, hence:
\[ \cos\left(\frac{\pi}{7}\right) + \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{3\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{5\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) = 0 \]

### (iii) Find \( \cos\left(\frac{\pi}{11}\right) + \cos\left(\frac{3\pi}{11}\right) + \cos\left(\frac{5\pi}{11}\right) + \cos\left(\frac{7\pi}{11}\right) + \cos\left(\frac{9\pi}{11}\right) \)

1. **Use the identity for the sum of cosines**:
We can apply the same approach as before:
\[ \cos\left(\frac{\pi}{11}\right) + \cos\left(\frac{10\pi}{11}\right) = 0 \]
\[ \cos\left(\frac{3\pi}{11}\right) + \cos\left(\frac{8\pi}{11}\right) = 0 \]
\[ \cos\left(\frac{5\pi}{11}\right) + \cos\left(\frac{6\pi}{11}\right) = 0 \]

2. **Sum up the pairs**:
All pairs sum to zero, hence:
\[ \cos\left(\frac{\pi}{11}\right) + \cos\left(\frac{3\pi}{11}\right) + \cos\left(\frac{5\pi}{11}\right) + \cos\left(\frac{7\pi}{11}\right) + \cos\left(\frac{9\pi}{11}\right) = 0 \]

### Final Results:
1. \( \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) = \cos\left(\frac{4\pi}{7}\right) \)
2. \( \cos\left(\frac{\pi}{7}\right) + \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{3\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{5\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) = 0 \)
3. \( \cos\left(\frac{\pi}{11}\right) + \cos\left(\frac{3\pi}{11}\right) + \cos\left(\frac{5\pi}{11}\right) + \cos\left(\frac{7\pi}{11}\right) + \cos\left(\frac{9\pi}{11}\right) = 0 \)