The general solution of the equation `8 cos x cos 2x cos 4x = sin 6x//sin x` is

Options

A. `x=(n pi//7)+(pi//21), AA n in Z`
B. `x=(2pi//7)+(pi//14), AA n in Z`
C. `x=(n pi//7)+(pi//14), AA n in Z`
D. `x=(n pi)+(pi//14), AA n in Z`

Explanation

To solve the equation \( 8 \cos x \cos 2x \cos 4x = \frac{\sin 6x}{\sin x} \), we will follow a systematic approach.

### Step-by-Step Solution:

1. **Rewrite the Equation:**
Start with the given equation:
\[ 8 \cos x \cos 2x \cos 4x = \frac{\sin 6x}{\sin x} \]

2. **Use Trigonometric Identities:**
We can use the identity \( 2 \cos A \sin B = \sin(A + B) - \sin(A - B) \). We will express the left-hand side using the product-to-sum identities.
\[ 8 \cos x \cos 2x \cos 4x = 4 \cdot 2 \cos x \cos 2x \cos 4x \]
We know that \( 2 \cos A \sin B = \sin(A + B) - \sin(A - B) \).

3. **Express \( \sin 6x \) using identities:**
We can express \( \sin 6x \) in terms of \( \sin x \):
\[ 8 \cos x \cos 2x \cos 4x = \sin 6x \cdot \frac{1}{\sin x} \]

4. **Simplify the Equation:**
Rearranging gives:
\[ 8 \cos x \cos 2x \cos 4x \sin x = \sin 6x \]

5. **Use the Identity for Sine:**
We can use the sine subtraction formula:
\[ \sin 6x - \sin 8x = 0 \]
This can be expressed as:
\[ 2 \cos\left(\frac{6x + 8x}{2}\right) \sin\left(\frac{8x - 6x}{2}\right) = 0 \]
Which simplifies to:
\[ 2 \cos(7x) \sin(x) = 0 \]

6. **Set Each Factor to Zero:**
This gives us two cases to consider:
- \( \cos(7x) = 0 \)
- \( \sin(x) = 0 \)

7. **Solve \( \cos(7x) = 0 \):**
The general solution for \( \cos(7x) = 0 \) is:
\[ 7x = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
Thus:
\[ x = \frac{\pi}{14} + \frac{n\pi}{7} \]

8. **Solve \( \sin(x) = 0 \):**
The general solution for \( \sin(x) = 0 \) is:
\[ x = k\pi \quad (k \in \mathbb{Z}) \]

9. **Combine Solutions:**
Since \( \sin x \) cannot be zero in the original equation (as it would make the denominator zero), we discard this case. Therefore, the only valid solution is:
\[ x = \frac{\pi}{14} + \frac{n\pi}{7} \quad (n \in \mathbb{Z}) \]

### Final Answer:
The general solution of the equation is:
\[ x = \frac{\pi}{14} + \frac{n\pi}{7}, \quad n \in \mathbb{Z} \]