Find the exact value of each of the following under the given conditions. \( \sin \alpha=\frac{7}{25}, 0<\alpha<\frac{\pi}{2} ; \cos \beta=\frac{8 \sqrt{89}}{89},-\frac{\pi}{2}<\beta<0 \) \( \begin{array}{llll}\text { (a) } \sin (\alpha+\beta) & \text { (b) } \cos (\alpha+\beta) & \text { (c) } \sin (\alpha-\beta) & \text { (d) } \tan (\alpha-\beta)\end{array} \) \( \begin{array}{ll}\text { (a) } \sin (\alpha+\beta)=-\frac{64 \sqrt{89}}{2225} & \text { (Type an exact answer using radicals as needed. Rationalize all denominators. Use integers or fractions for any numbers in the expression. Simplify your answer.) } \\ \text { (b) } \cos (\alpha+\beta)=\square & \text { (Type an exact answer using radicals as needed. Rationalize all denominators. Use integers or fractions for any numbers in the expression. Simplify your answer.) }\end{array} \)

To find the exact values, we start by determining the necessary trigonometric values based on the given information. 1. **Finding \( \cos \alpha \):** Using the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \): \[ \sin^2 \alpha = \left(\frac{7}{25}\right)^2 = \frac{49}{625} \] \[ \cos^2 \alpha = 1 - \frac{49}{625} = \frac{576}{625} \] \[ \cos \alpha = \sqrt{\frac{576}{625}} = \frac{24}{25} \] 2. **Finding \( \sin \beta \):** Given \( \cos \beta = \frac{8\sqrt{89}}{89} \), we also use the Pythagorean identity: \[ \cos^2 \beta + \sin^2 \beta = 1 \] \[ \sin^2 \beta = 1 - \left(\frac{8\sqrt{89}}{89}\right)^2 = 1 - \frac{64 \cdot 89}{7921} = 1 - \frac{5696}{7921} = \frac{2225}{7921} \] \[ \sin \beta = -\sqrt{\frac{2225}{7921}} = -\frac{\sqrt{2225}}{89} \] (Note: \( \sin \beta \) is negative because \( \beta \) is in the fourth quadrant.) 3. **Calculating \( \sin(\alpha + \beta) \):** Using the sine addition formula: \[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \] Substituting the values: \[ \sin(\alpha + \beta) = \frac{7}{25} \cdot \frac{8\sqrt{89}}{89} + \frac{24}{25} \cdot \left(-\frac{\sqrt{2225}}{89}\right) \] \[ = \frac{56\sqrt{89}}{1775} - \frac{24\sqrt{2225}}{2225} \] Common denominator for the two terms: \[ = \frac{56\sqrt{89} - 24\sqrt{2225}}{1775} \] 4. **Calculating \( \cos(\alpha + \beta) \):** Using the cosine addition formula: \[ \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \] \[ = \frac{24}{25} \cdot \frac{8\sqrt{89}}{89} - \frac{7}{25} \cdot \left(-\frac{\sqrt{2225}}{89}\right) \] \[ = \frac{192\sqrt{89}}{2225} + \frac{7\sqrt{2225}}{2225} \] \[ = \frac{192\sqrt{89} + 7\sqrt{2225}}{2225} \] So, the answers are: \[ \text{(a)} \sin(\alpha + \beta) = \frac{56\sqrt{89} - 24\sqrt{2225}}{1775} \] \[ \text{(b)} \cos(\alpha + \beta) = \frac{192\sqrt{89} + 7\sqrt{2225}}{2225} \]