If `tan(A+B)=pand tan(A-B)=q,` then the value of tan2A, is

Options

A. `(p+q)/(p-q)`
B. `(p-q)/(1+pq)`
C. `(1+pq)/(1+pq)`
D. `(p+q)/(1-pq)`

Answer

To find the value of \( \tan 2A \) given that \( \tan(A+B) = p \) and \( \tan(A-B) = q \), we can use the tangent addition and subtraction formulas.

### Step-by-Step Solution:

1. **Recall the formula for \( \tan 2A \)**:
\[ \tan 2A = \frac{\tan(A+B) + \tan(A-B)}{1 - \tan(A+B) \tan(A-B)} \]

2. **Substitute the given values**:
Since we know that \( \tan(A+B) = p \) and \( \tan(A-B) = q \), we can substitute these values into the formula:
\[ \tan 2A = \frac{p + q}{1 - pq} \]

3. **Final expression**:
Thus, the value of \( \tan 2A \) is:
\[ \tan 2A = \frac{p + q}{1 - pq} \]

### Conclusion:
The value of \( \tan 2A \) is \( \frac{p + q}{1 - pq} \).