Find the exact value of the expressions \( \cos (\alpha+\beta), \sin (\alpha+\beta) \) and \( \tan (\alpha+\beta) \) under the following conditions: \[ \sin (\alpha)=\frac{15}{17}, \alpha \text { lies in quadrant I, and } \sin (\beta)=\frac{4}{5}, \beta \text { lies in quadrant II. } \]

To find the exact values of \( \cos(\alpha+\beta) \), \( \sin(\alpha+\beta) \), and \( \tan(\alpha+\beta) \), we first need to find \( \cos(\alpha) \) and \( \cos(\beta) \). Given \( \sin(\alpha) = \frac{15}{17} \) and \( \alpha \) is in quadrant I, we can use the Pythagorean identity: \[ \cos^2(\alpha) + \sin^2(\alpha) = 1 \implies \cos^2(\alpha) + \left(\frac{15}{17}\right)^2 = 1, \] \[ \cos^2(\alpha) + \frac{225}{289} = 1 \implies \cos^2(\alpha) = 1 - \frac{225}{289} = \frac{64}{289} \implies \cos(\alpha) = \frac{8}{17}. \] Next, for \( \beta \), we have \( \sin(\beta) = \frac{4}{5} \) and \( \beta \) is in quadrant II. Since cosine is negative in quadrant II, we find: \[ \cos^2(\beta) + \sin^2(\beta) = 1 \implies \cos^2(\beta) + \left(\frac{4}{5}\right)^2 = 1, \] \[ \cos^2(\beta) + \frac{16}{25} = 1 \implies \cos^2(\beta) = 1 - \frac{16}{25} = \frac{9}{25} \implies \cos(\beta) = -\frac{3}{5}. \] Now we can apply the angle sum formulas: 1. **Finding \( \sin(\alpha + \beta) \)**: \[ \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta). \] Substituting the values: \[ \sin(\alpha + \beta) = \left(\frac{15}{17}\right)\left(-\frac{3}{5}\right) + \left(\frac{8}{17}\right)\left(\frac{4}{5}\right), \] \[ = -\frac{45}{85} + \frac{32}{85} = -\frac{13}{85}. \] 2. **Finding \( \cos(\alpha + \beta) \)**: \[ \cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta). \] Substituting the values: \[ \cos(\alpha + \beta) = \left(\frac{8}{17}\right)\left(-\frac{3}{5}\right) - \left(\frac{15}{17}\right)\left(\frac{4}{5}\right), \] \[ = -\frac{24}{85} - \frac{60}{85} = -\frac{84}{85}. \] 3. **Finding \( \tan(\alpha + \beta) \)**: \[ \tan(\alpha + \beta) = \frac{\sin(\alpha + \beta)}{\cos(\alpha + \beta)}. \] Substituting the values: \[ \tan(\alpha + \beta) = \frac{-\frac{13}{85}}{-\frac{84}{85}} = \frac{13}{84}. \] Thus, the exact values are: \[ \sin(\alpha + \beta) = -\frac{13}{85}, \quad \cos(\alpha + \beta) = -\frac{84}{85}, \quad \tan(\alpha + \beta) = \frac{13}{84}. \]