1. **State the problem:** We need to verify or simplify the equation $$\frac{1 + \tan y}{1 + \cot y} = \frac{\sec y}{\csc y}$$. 2. **Recall the definitions:** - $$\tan y = \frac{\sin y}{\cos y}$$ - $$\cot y = \frac{\cos y}{\sin y}$$ - $$\sec y = \frac{1}{\cos y}$$ - $$\csc y = \frac{1}{\sin y}$$ 3. **Rewrite the left side using definitions:** $$\frac{1 + \tan y}{1 + \cot y} = \frac{1 + \frac{\sin y}{\cos y}}{1 + \frac{\cos y}{\sin y}}$$ 4. **Find common denominators inside numerator and denominator:** Numerator: $$1 + \frac{\sin y}{\cos y} = \frac{\cos y}{\cos y} + \frac{\sin y}{\cos y} = \frac{\cos y + \sin y}{\cos y}$$ Denominator: $$1 + \frac{\cos y}{\sin y} = \frac{\sin y}{\sin y} + \frac{\cos y}{\sin y} = \frac{\sin y + \cos y}{\sin y}$$ 5. **Rewrite the entire fraction:** $$\frac{\frac{\cos y + \sin y}{\cos y}}{\frac{\sin y + \cos y}{\sin y}}$$ 6. **Divide by a fraction by multiplying by its reciprocal:** $$= \frac{\cos y + \sin y}{\cos y} \times \frac{\sin y}{\sin y + \cos y}$$ 7. **Cancel common factors:** Since $$\cos y + \sin y = \sin y + \cos y$$, they cancel: $$= \cancel{\frac{\cos y + \sin y}{\cos y}} \times \cancel{\frac{\sin y}{\sin y + \cos y}} = \frac{\sin y}{\cos y}$$ 8. **Simplify the left side:** $$\frac{1 + \tan y}{1 + \cot y} = \frac{\sin y}{\cos y} = \tan y$$ 9. **Simplify the right side:** $$\frac{\sec y}{\csc y} = \frac{\frac{1}{\cos y}}{\frac{1}{\sin y}} = \frac{1}{\cos y} \times \frac{\sin y}{1} = \frac{\sin y}{\cos y} = \tan y$$ 10. **Conclusion:** Both sides simplify to $$\tan y$$, so the equation is true. **Final answer:** $$\frac{1 + \tan y}{1 + \cot y} = \frac{\sec y}{\csc y} = \tan y$$