Rewrite \( \frac{\cos \theta+\sin \theta}{\sin \theta}+\frac{\sin \theta-\cos \theta}{\cos \theta} \) over a common denominator. Type your answer in terms of sine and/or cosine. \( \frac{\cos \theta+\sin \theta}{\sin \theta}+\frac{\sin \theta-\cos \theta}{\cos \theta}=\square \) (Simplify your answer.)

To combine the expression \( \frac{\cos \theta+\sin \theta}{\sin \theta}+\frac{\sin \theta-\cos \theta}{\cos \theta} \) over a common denominator, we first identify the denominators, which are \( \sin \theta \) and \( \cos \theta \). The common denominator will be \( \sin \theta \cos \theta \). Now, we rewrite each fraction with the common denominator: 1. For the first fraction: \[ \frac{\cos \theta + \sin \theta}{\sin \theta} = \frac{(\cos \theta + \sin \theta) \cos \theta}{\sin \theta \cos \theta} = \frac{\cos^2 \theta + \sin \theta \cos \theta}{\sin \theta \cos \theta} \] 2. For the second fraction: \[ \frac{\sin \theta - \cos \theta}{\cos \theta} = \frac{(\sin \theta - \cos \theta) \sin \theta}{\sin \theta \cos \theta} = \frac{\sin^2 \theta - \sin \theta \cos \theta}{\sin \theta \cos \theta} \] Now we can combine the two fractions: \[ \frac{\cos^2 \theta + \sin \theta \cos \theta}{\sin \theta \cos \theta} + \frac{\sin^2 \theta - \sin \theta \cos \theta}{\sin \theta \cos \theta} = \frac{\cos^2 \theta + \sin \theta \cos \theta + \sin^2 \theta - \sin \theta \cos \theta}{\sin \theta \cos \theta} \] Notice that \( \sin \theta \cos \theta \) cancels out: \[ \cos^2 \theta + \sin^2 \theta = 1 \] Thus, we have: \[ \frac{1}{\sin \theta \cos \theta} \] Finally, we can express this in terms of sine and cosine: \[ \frac{1}{\sin \theta \cos \theta} = \frac{2}{\sin(2\theta)} \] So the simplified expression is: \[ \frac{1}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta} \] Thus, the final answer is: \[ \frac{1}{\sin \theta \cos \theta} \]