statement-I- `cos^3alpha + cos^3(alpha+2pi/3) + cos^3(alpha+4pi/3) = ``3cosalpha cos(alpha+2pi/3)cos(alpha+4pi/3) Because` Statement-II - `if a+b+c=0 iff a^3+b^3+c^3=3abc`

Options

A. A
B. B
C. C
D. D

Answer

To solve the problem, we need to prove the statement:

**Statement I:**
\[ \cos^3 \alpha + \cos^3 \left( \alpha + \frac{2\pi}{3} \right) + \cos^3 \left( \alpha + \frac{4\pi}{3} \right) = 3 \cos \alpha \cos \left( \alpha + \frac{2\pi}{3} \right) \cos \left( \alpha + \frac{4\pi}{3} \right) \]

**Statement II:**
If \( a + b + c = 0 \), then \( a^3 + b^3 + c^3 = 3abc \).

### Step-by-Step Solution

1. **Identify \( a, b, c \):**
Let:
\[ a = \cos \alpha, \quad b = \cos \left( \alpha + \frac{2\pi}{3} \right), \quad c = \cos \left( \alpha + \frac{4\pi}{3} \right) \]

2. **Sum of angles:**
We know that:
\[ b + c = \cos \left( \alpha + \frac{2\pi}{3} \right) + \cos \left( \alpha + \frac{4\pi}{3} \right) \]

Using the cosine addition formula:
\[ \cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \]

Here, \( A = \alpha + \frac{2\pi}{3} \) and \( B = \alpha + \frac{4\pi}{3} \):
\[ b + c = 2 \cos \left( \alpha + \frac{\pi}{3} \right) \cos \left( \frac{-\pi}{3} \right) = 2 \cos \left( \alpha + \frac{\pi}{3} \right) \left(-\frac{1}{2}\right) = -\cos \left( \alpha + \frac{\pi}{3} \right) \]

3. **Check if \( a + b + c = 0 \):**
Now we can write:
\[ a + b + c = \cos \alpha + \cos \left( \alpha + \frac{2\pi}{3} \right) + \cos \left( \alpha + \frac{4\pi}{3} \right) = 0 \]

This confirms that \( a + b + c = 0 \).

4. **Use Statement II:**
According to Statement II, since \( a + b + c = 0 \):
\[ a^3 + b^3 + c^3 = 3abc \]

Therefore:
\[ \cos^3 \alpha + \cos^3 \left( \alpha + \frac{2\pi}{3} \right) + \cos^3 \left( \alpha + \frac{4\pi}{3} \right) = 3 \cos \alpha \cos \left( \alpha + \frac{2\pi}{3} \right) \cos \left( \alpha + \frac{4\pi}{3} \right) \]

5. **Conclusion:**
Thus, we have proved Statement I using Statement II.

### Final Answer:
Both Statement I and Statement II are correct, and Statement II is the correct explanation for Statement I.