1. **State the problem:** Verify the trigonometric identity: $$\frac{1 + \sin \theta}{\cos \theta} + \frac{\cos \theta}{1 + \sin \theta} = 2 \sec \theta$$ 2. **Recall important formulas and rules:** - $\sec \theta = \frac{1}{\cos \theta}$ - Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$ - To verify an identity, simplify one side to see if it equals the other. 3. **Start with the left-hand side (LHS):** $$\frac{1 + \sin \theta}{\cos \theta} + \frac{\cos \theta}{1 + \sin \theta}$$ 4. **Find a common denominator:** The common denominator is $\cos \theta (1 + \sin \theta)$. Rewrite each term: $$\frac{(1 + \sin \theta)^2}{\cos \theta (1 + \sin \theta)} + \frac{\cos^2 \theta}{\cos \theta (1 + \sin \theta)}$$ 5. **Combine the fractions:** $$\frac{(1 + \sin \theta)^2 + \cos^2 \theta}{\cos \theta (1 + \sin \theta)}$$ 6. **Expand numerator:** $$(1 + \sin \theta)^2 = 1 + 2 \sin \theta + \sin^2 \theta$$ So numerator becomes: $$1 + 2 \sin \theta + \sin^2 \theta + \cos^2 \theta$$ 7. **Use Pythagorean identity:** Replace $\sin^2 \theta + \cos^2 \theta$ with 1: $$1 + 2 \sin \theta + 1 = 2 + 2 \sin \theta$$ 8. **Simplify numerator:** $$2 + 2 \sin \theta = 2(1 + \sin \theta)$$ 9. **Substitute back into fraction:** $$\frac{2(1 + \sin \theta)}{\cos \theta (1 + \sin \theta)}$$ 10. **Cancel common factor $(1 + \sin \theta)$:** $$\frac{2}{\cos \theta}$$ 11. **Rewrite using $\sec \theta$:** $$2 \sec \theta$$ 12. **Conclusion:** LHS simplifies to RHS, so the identity is verified: $$\frac{1 + \sin \theta}{\cos \theta} + \frac{\cos \theta}{1 + \sin \theta} = 2 \sec \theta$$