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To solve for \( A \) and \( B \), we use trigonometric identities and the equation given:

\[
\cos\left(\frac{3\pi}{7}\right) \cos\left(\frac{2\pi}{21}\right) + \sin\left(\frac{3\pi}{7}\right) \sin\left(\frac{2\pi}{21}\right) = \cos\left(\frac{\pi}{A}\right) = \frac{B}{2}
\]

### Step 1: Apply the Cosine Addition Formula
The expression on the left side can be simplified using the cosine addition formula:
\[
\cos(x)\cos(y) + \sin(x)\sin(y) = \cos(x - y)
\]
Substituting \( x = \frac{3\pi}{7} \) and \( y = \frac{2\pi}{21} \):
\[
\cos\left(\frac{3\pi}{7}\right) \cos\left(\frac{2\pi}{21}\right) + \sin\left(\frac{3\pi}{7}\right) \sin\left(\frac{2\pi}{21}\right) = \cos\left(\frac{3\pi}{7} - \frac{2\pi}{21}\right)
\]

### Step 2: Simplify the Argument
To simplify \( \frac{3\pi}{7} - \frac{2\pi}{21} \), find a common denominator:
\[
\frac{3\pi}{7} - \frac{2\pi}{21} = \frac{9\pi}{21} - \frac{2\pi}{21} = \frac{7\pi}{21} = \frac{\pi}{3}
\]

So, the equation becomes:
\[
\cos\left(\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{A}\right)
\]

### Step 3: Solve for \( A \)
Since \( \cos\left(\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{A}\right) \), it follows that:
\[
A = 3
\]

### Step 4: Find \( B \)
We know that \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \). Therefore:
\[
\frac{B}{2} = \frac{1}{2}
\]

Solving for \( B \):
\[
B = 1
\]

### Final Solution
\[
\boxed{A = 3, B = 1}
\]

Would you like a more detailed explanation, or do you have any questions?

Here are some related questions for further exploration:
1. What other trigonometric identities can be used to simplify similar expressions?
2. How does the cosine subtraction formula differ from the addition formula?
3. Can the values of \( A \) and \( B \) change if we choose different trigonometric identities?
4. How can we verify the solution using another method, such as converting to radians?
5. What is the significance of the angles \( \frac{3\pi}{7} \) and \( \frac{2\pi}{21} \) in trigonometric simplifications?

**Tip:** Always check if the angles given can be simplified using a common denominator to make the calculations easier.

What are the values of A and B in the trigonometric identity?

Learn how to solve for A and B in the given trigonometric equation using the cosine addition formula. This problem covers key trigonometric concepts and identities, suitable for students in Grades 11-12.

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