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.