1. **State the problem:** Simplify the expression $$\frac{\sec x + \sin x \sec x}{\cos x}$$. 2. **Recall definitions and formulas:** - Recall that $$\sec x = \frac{1}{\cos x}$$. - The goal is to simplify the expression using trigonometric identities. 3. **Rewrite the numerator:** $$\sec x + \sin x \sec x = \sec x (1 + \sin x)$$ 4. **Substitute $$\sec x = \frac{1}{\cos x}$$:** $$\frac{1}{\cos x} (1 + \sin x)$$ 5. **Rewrite the entire expression:** $$\frac{\sec x + \sin x \sec x}{\cos x} = \frac{\frac{1}{\cos x} (1 + \sin x)}{\cos x}$$ 6. **Simplify the complex fraction:** $$= \frac{1 + \sin x}{\cos x} \times \frac{1}{\cos x} = \frac{1 + \sin x}{\cancel{\cos x}} \times \frac{1}{\cancel{\cos x}}$$ 7. **Combine the denominators:** $$= \frac{1 + \sin x}{\cos^2 x}$$ 8. **Final simplified form:** $$\boxed{\frac{1 + \sin x}{\cos^2 x}}$$ This is the simplified expression.