If `alpha and beta` be between `0 and (pi)/(2)and if cos(alpha+beta)=(12)/(13) and sin(alpha-beta)=3/5,` then sin 2 `alpha` is equal to

Options

A. `64//65`
B. `56//65`
C. 0
D. `16//15`

Answer

To solve the problem step by step, we will use the given information and trigonometric identities.

### Step 1: Understand the given information
We have:
- \( \cos(\alpha + \beta) = \frac{12}{13} \)
- \( \sin(\alpha - \beta) = \frac{3}{5} \)

### Step 2: Find \( \sin(\alpha + \beta) \)
Using the Pythagorean identity, we can find \( \sin(\alpha + \beta) \):
\[ \sin(\alpha + \beta) = \sqrt{1 - \cos^2(\alpha + \beta)} = \sqrt{1 - \left(\frac{12}{13}\right)^2} \]
Calculating this:
\[ \sin(\alpha + \beta) = \sqrt{1 - \frac{144}{169}} = \sqrt{\frac{25}{169}} = \frac{5}{13} \]

### Step 3: Find \( \cos(\alpha - \beta) \)
Using the Pythagorean identity for \( \sin(\alpha - \beta) \):
\[ \cos(\alpha - \beta) = \sqrt{1 - \sin^2(\alpha - \beta)} = \sqrt{1 - \left(\frac{3}{5}\right)^2} \]
Calculating this:
\[ \cos(\alpha - \beta) = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5} \]

### Step 4: Use the sine addition formula
We know:
\[ \sin(2\alpha) = \sin((\alpha + \beta) + (\alpha - \beta)) = \sin(\alpha + \beta)\cos(\alpha - \beta) + \cos(\alpha + \beta)\sin(\alpha - \beta) \]
Substituting the values we have:
\[ \sin(2\alpha) = \left(\frac{5}{13}\right) \left(\frac{4}{5}\right) + \left(\frac{12}{13}\right) \left(\frac{3}{5}\right) \]

### Step 5: Calculate \( \sin(2\alpha) \)
Calculating each term:
\[ \sin(2\alpha) = \frac{5 \cdot 4}{13 \cdot 5} + \frac{12 \cdot 3}{13 \cdot 5} = \frac{20}{65} + \frac{36}{65} = \frac{56}{65} \]

### Conclusion
Thus, the value of \( \sin(2\alpha) \) is:
\[ \sin(2\alpha) = \frac{56}{65} \]