If sin `alpha=(15)/(17)` and cos `beta=(12)/(13)` find the values of:
(i)`"sin"(alpha+beta)`
(ii)` cos(alpha-beta)`
(iii)` "tan"(alpha+beta)`

Answer

To solve the problem, we will follow these steps:

### Given:
- \( \sin \alpha = \frac{15}{17} \)
- \( \cos \beta = \frac{12}{13} \)

### Step 1: Find \( \cos \alpha \)
Using the identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \):
\[ \cos^2 \alpha = 1 - \sin^2 \alpha \]
Substituting the value of \( \sin \alpha \):
\[ \cos^2 \alpha = 1 - \left(\frac{15}{17}\right)^2 = 1 - \frac{225}{289} = \frac{64}{289} \]
Taking the square root:
\[ \cos \alpha = \frac{8}{17} \]

### Step 2: Find \( \sin \beta \)
Using the identity \( \sin^2 \beta + \cos^2 \beta = 1 \):
\[ \sin^2 \beta = 1 - \cos^2 \beta \]
Substituting the value of \( \cos \beta \):
\[ \sin^2 \beta = 1 - \left(\frac{12}{13}\right)^2 = 1 - \frac{144}{169} = \frac{25}{169} \]
Taking the square root:
\[ \sin \beta = \frac{5}{13} \]

### Step 3: Calculate \( \sin(\alpha + \beta) \)
Using the formula:
\[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \]
Substituting the known values:
\[ \sin(\alpha + \beta) = \left(\frac{15}{17}\right) \left(\frac{12}{13}\right) + \left(\frac{8}{17}\right) \left(\frac{5}{13}\right) \]
Calculating:
\[ = \frac{180}{221} + \frac{40}{221} = \frac{220}{221} \]

### Step 4: Calculate \( \cos(\alpha - \beta) \)
Using the formula:
\[ \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \]
Substituting the known values:
\[ \cos(\alpha - \beta) = \left(\frac{8}{17}\right) \left(\frac{12}{13}\right) + \left(\frac{15}{17}\right) \left(\frac{5}{13}\right) \]
Calculating:
\[ = \frac{96}{221} + \frac{75}{221} = \frac{171}{221} \]

### Step 5: Calculate \( \tan(\alpha + \beta) \)
Using the formula:
\[ \tan(\alpha + \beta) = \frac{\sin(\alpha + \beta)}{\cos(\alpha + \beta)} \]
First, we need to find \( \cos(\alpha + \beta) \):
Using the identity \( \cos^2(\alpha + \beta) = 1 - \sin^2(\alpha + \beta) \):
\[ \cos^2(\alpha + \beta) = 1 - \left(\frac{220}{221}\right)^2 \]
Calculating:
\[ = 1 - \frac{48400}{48841} = \frac{441}{48841} \]
Taking the square root:
\[ \cos(\alpha + \beta) = \frac{21}{221} \]
Now substituting into the tangent formula:
\[ \tan(\alpha + \beta) = \frac{\frac{220}{221}}{\frac{21}{221}} = \frac{220}{21} \]

### Final Answers:
1. \( \sin(\alpha + \beta) = \frac{220}{221} \)
2. \( \cos(\alpha - \beta) = \frac{171}{221} \)
3. \( \tan(\alpha + \beta) = \frac{220}{21} \)