Given that \( \cos \theta=\frac{8}{17} \cdot \operatorname{col} \) culate \( \frac{\sin ^{2} \theta+\cos ^{2} \theta}{\sin \theta+\cos \theta} \) \( \frac{\cos \theta}{\sin \theta} \)

To solve the problem, we start with the known condition: 1. \( \cos \theta = \frac{8}{17} \) We need to calculate two expressions: 1. \( \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta + \cos \theta} \) 2. \( \frac{\cos \theta}{\sin \theta} \) ### Step 1: Calculate \( \sin \theta \) Using the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Substituting \( \cos \theta \): \[ \sin^2 \theta + \left(\frac{8}{17}\right)^2 = 1 \] Calculating \( \left(\frac{8}{17}\right)^2 \): \[ \left(\frac{8}{17}\right)^2 = \frac{64}{289} \] Now substituting back into the identity: \[ \sin^2 \theta + \frac{64}{289} = 1 \] To isolate \( \sin^2 \theta \): \[ \sin^2 \theta = 1 - \frac{64}{289} = \frac{289}{289} - \frac{64}{289} = \frac{225}{289} \] Taking the square root to find \( \sin \theta \): \[ \sin \theta = \sqrt{\frac{225}{289}} = \frac{15}{17} \] ### Step 2: Calculate the first expression Now we can calculate: \[ \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta + \cos \theta} \] We already know \( \sin^2 \theta + \cos^2 \theta = 1 \), so we need to find \( \sin \theta + \cos \theta \): \[ \sin \theta + \cos \theta = \frac{15}{17} + \frac{8}{17} = \frac{23}{17} \] Now substituting into the expression: \[ \frac{1}{\frac{23}{17}} = \frac{17}{23} \] ### Step 3: Calculate the second expression Now we calculate: \[ \frac{\cos \theta}{\sin \theta} = \frac{\frac{8}{17}}{\frac{15}{17}} = \frac{8}{15} \] ### Final Results Thus, the final results are: 1. \( \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta + \cos \theta} = \frac{17}{23} \) 2. \( \frac{\cos \theta}{\sin \theta} = \frac{8}{15} \)