Statement I `sin((2pi)/(7))+sin((4pi)/(7))+sin((8pi)/(7))=- 1/2`.
Statement II `cos. (2pi)/(7)+I sin . (2pi)/(7)` is complex 7th root of unity.

Options

A. Both Statement I and Statement II are individually true and R is the correct explanation of Statement I.
B. Both Statement I and Statement II are individually true but Statement II is not the correct explanaton of Statement I.
C. Statement I is true but Statement II is false.
D. Statement I is false but Statement II is true.

Answer

To solve the problem, we need to evaluate the two statements:

**Statement I:** \( \sin\left(\frac{2\pi}{7}\right) + \sin\left(\frac{4\pi}{7}\right) + \sin\left(\frac{8\pi}{7}\right) = -\frac{1}{2} \)

**Statement II:** \( \cos\left(\frac{2\pi}{7}\right) + i \sin\left(\frac{2\pi}{7}\right) \) is a complex 7th root of unity.

### Step 1: Evaluate Statement II

The complex 7th roots of unity are given by:
\[ 1, e^{i\frac{2\pi}{7}}, e^{i\frac{4\pi}{7}}, e^{i\frac{6\pi}{7}}, e^{i\frac{8\pi}{7}}, e^{i\frac{10\pi}{7}}, e^{i\frac{12\pi}{7}} \]
We can express \( e^{i\frac{2\pi}{7}} \) using Euler's formula:
\[ e^{i\frac{2\pi}{7}} = \cos\left(\frac{2\pi}{7}\right) + i \sin\left(\frac{2\pi}{7}\right) \]
Thus, Statement II is true since \( e^{i\frac{2\pi}{7}} \) is indeed one of the 7th roots of unity.

### Step 2: Evaluate Statement I

We need to check if:
\[ \sin\left(\frac{2\pi}{7}\right) + \sin\left(\frac{4\pi}{7}\right) + \sin\left(\frac{8\pi}{7}\right) = -\frac{1}{2} \]

Using the identity \( \sin(\pi - x) = \sin(x) \), we can rewrite:
\[ \sin\left(\frac{8\pi}{7}\right) = \sin\left(\pi - \frac{6\pi}{7}\right) = \sin\left(\frac{6\pi}{7}\right) \]
Thus, we have:
\[ \sin\left(\frac{2\pi}{7}\right) + \sin\left(\frac{4\pi}{7}\right) + \sin\left(\frac{6\pi}{7}\right) \]

Now, we know from the property of sine functions that:
\[ \sin\left(\frac{2\pi}{7}\right) + \sin\left(\frac{4\pi}{7}\right) + \sin\left(\frac{6\pi}{7}\right) + \sin\left(\frac{8\pi}{7}\right) + \sin\left(\frac{10\pi}{7}\right) + \sin\left(\frac{12\pi}{7}\right) = 0 \]
This means:
\[ \sin\left(\frac{2\pi}{7}\right) + \sin\left(\frac{4\pi}{7}\right) + \sin\left(\frac{6\pi}{7}\right) = -\left(\sin\left(\frac{8\pi}{7}\right) + \sin\left(\frac{10\pi}{7}\right) + \sin\left(\frac{12\pi}{7}\right)\right) \]
Since \( \sin\left(\frac{8\pi}{7}\right) = -\sin\left(\frac{6\pi}{7}\right) \), we can conclude:
\[ \sin\left(\frac{2\pi}{7}\right) + \sin\left(\frac{4\pi}{7}\right) + \sin\left(\frac{8\pi}{7}\right) = 0 \]

Thus, Statement I is false because it claims the sum equals \(-\frac{1}{2}\), while we have shown it equals \(0\).

### Conclusion
- Statement I is **false**.
- Statement II is **true**.