1. **State the problem:** Simplify the expression $$\frac{\sin \theta}{1+\cos \theta} + \frac{1+\cos \theta}{\sin \theta} = 2 \csc \theta$$ and verify the equality. 2. **Recall important identities:** - $\csc \theta = \frac{1}{\sin \theta}$ - Use algebraic manipulation and trigonometric identities to simplify. 3. **Start with the left-hand side (LHS):** $$\frac{\sin \theta}{1+\cos \theta} + \frac{1+\cos \theta}{\sin \theta}$$ 4. **Find common denominator:** $$\frac{\sin^2 \theta}{(1+\cos \theta) \sin \theta} + \frac{(1+\cos \theta)^2}{(1+\cos \theta) \sin \theta} = \frac{\sin^2 \theta + (1+\cos \theta)^2}{(1+\cos \theta) \sin \theta}$$ 5. **Expand numerator:** $$\sin^2 \theta + (1 + 2\cos \theta + \cos^2 \theta) = \sin^2 \theta + 1 + 2\cos \theta + \cos^2 \theta$$ 6. **Use Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$:** $$1 + 1 + 2\cos \theta = 2 + 2\cos \theta = 2(1 + \cos \theta)$$ 7. **Substitute back into fraction:** $$\frac{2(1 + \cos \theta)}{(1+\cos \theta) \sin \theta}$$ 8. **Cancel common factor $1 + \cos \theta$:** $$\frac{\cancel{2(1 + \cos \theta)}}{\cancel{(1+\cos \theta)} \sin \theta} = \frac{2}{\sin \theta}$$ 9. **Rewrite using cosecant:** $$\frac{2}{\sin \theta} = 2 \csc \theta$$ 10. **Conclusion:** The left-hand side simplifies exactly to the right-hand side, so the equality holds. **Final answer:** $$\frac{\sin \theta}{1+\cos \theta} + \frac{1+\cos \theta}{\sin \theta} = 2 \csc \theta$$