1. **State the problem:** Prove that $$\frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1} = \frac{1 + \sin \theta}{\cos \theta}$$. 2. **Recall definitions:** - $$\tan \theta = \frac{\sin \theta}{\cos \theta}$$ - $$\sec \theta = \frac{1}{\cos \theta}$$ 3. **Rewrite the left-hand side (LHS) using these definitions:** $$\frac{\frac{\sin \theta}{\cos \theta} + \frac{1}{\cos \theta} - 1}{\frac{\sin \theta}{\cos \theta} - \frac{1}{\cos \theta} + 1}$$ 4. **Find a common denominator $$\cos \theta$$ in numerator and denominator:** $$\frac{\frac{\sin \theta + 1 - \cos \theta}{\cos \theta}}{\frac{\sin \theta - 1 + \cos \theta}{\cos \theta}}$$ 5. **Divide the fractions by multiplying numerator by reciprocal of denominator:** $$\frac{\sin \theta + 1 - \cos \theta}{\cos \theta} \times \frac{\cos \theta}{\sin \theta - 1 + \cos \theta} = \frac{\sin \theta + 1 - \cos \theta}{\sin \theta - 1 + \cos \theta}$$ 6. **Simplify numerator and denominator:** Rewrite denominator as $$\sin \theta + (\cos \theta - 1)$$. 7. **Notice numerator and denominator are similar except for signs:** Numerator: $$\sin \theta + 1 - \cos \theta$$ Denominator: $$\sin \theta + \cos \theta - 1$$ 8. **Add and subtract 1 in numerator and denominator to factor:** Rewrite numerator as $$ (1 + \sin \theta) - \cos \theta $$ Rewrite denominator as $$ (\sin \theta - 1) + \cos \theta $$ 9. **Multiply numerator and denominator by the conjugate of denominator's denominator to rationalize:** Multiply numerator and denominator by $$ (1 + \cos \theta) $$: $$\frac{(1 + \sin \theta - \cos \theta)(1 + \cos \theta)}{(\sin \theta + \cos \theta - 1)(1 + \cos \theta)}$$ 10. **Expand numerator:** $$ (1 + \sin \theta)(1 + \cos \theta) - \cos \theta (1 + \cos \theta) = (1 + \sin \theta)(1 + \cos \theta) - \cos \theta - \cos^2 \theta $$ 11. **Expand denominator:** $$ (\sin \theta + \cos \theta - 1)(1 + \cos \theta) = \sin \theta (1 + \cos \theta) + \cos \theta (1 + \cos \theta) - 1 (1 + \cos \theta) $$ 12. **Simplify numerator:** $$ (1 + \sin \theta)(1 + \cos \theta) - \cos \theta - \cos^2 \theta = 1 + \cos \theta + \sin \theta + \sin \theta \cos \theta - \cos \theta - \cos^2 \theta $$ Cancel $$+ \cos \theta$$ and $$- \cos \theta$$: $$ 1 + \sin \theta + \sin \theta \cos \theta - \cos^2 \theta $$ 13. **Simplify denominator:** $$ \sin \theta + \sin \theta \cos \theta + \cos \theta + \cos^2 \theta - 1 - \cos \theta = \sin \theta + \sin \theta \cos \theta + \cos^2 \theta - 1 $$ 14. **Rewrite numerator and denominator:** Numerator: $$ 1 + \sin \theta + \sin \theta \cos \theta - \cos^2 \theta $$ Denominator: $$ \sin \theta + \sin \theta \cos \theta + \cos^2 \theta - 1 $$ 15. **Use Pythagorean identity $$\sin^2 \theta + \cos^2 \theta = 1$$ to rewrite $$1 - \cos^2 \theta = \sin^2 \theta$$:** Rewrite numerator: $$ 1 + \sin \theta + \sin \theta \cos \theta - \cos^2 \theta = (1 - \cos^2 \theta) + \sin \theta + \sin \theta \cos \theta = \sin^2 \theta + \sin \theta + \sin \theta \cos \theta $$ Rewrite denominator: $$ \sin \theta + \sin \theta \cos \theta + \cos^2 \theta - 1 = \sin \theta + \sin \theta \cos \theta - (1 - \cos^2 \theta) = \sin \theta + \sin \theta \cos \theta - \sin^2 \theta $$ 16. **Factor numerator and denominator:** Numerator: $$ \sin^2 \theta + \sin \theta + \sin \theta \cos \theta = \sin \theta (\sin \theta + 1 + \cos \theta) $$ Denominator: $$ \sin \theta + \sin \theta \cos \theta - \sin^2 \theta = \sin \theta (1 + \cos \theta - \sin \theta) $$ 17. **Cancel common factor $$\sin \theta$$ (assuming $$\sin \theta \neq 0$$):** $$ \frac{\sin \theta + 1 + \cos \theta}{1 + \cos \theta - \sin \theta} $$ 18. **Rewrite denominator:** $$ 1 + \cos \theta - \sin \theta = (1 + \cos \theta) - \sin \theta $$ 19. **Notice numerator and denominator are conjugates:** Numerator: $$ \sin \theta + 1 + \cos \theta = (1 + \sin \theta) + \cos \theta $$ Denominator: $$ (1 + \cos \theta) - \sin \theta $$ 20. **Multiply numerator and denominator by $$ (1 + \sin \theta) + \cos \theta $$ to rationalize denominator:** $$ \frac{(1 + \sin \theta) + \cos \theta}{(1 + \cos \theta) - \sin \theta} \times \frac{(1 + \sin \theta) + \cos \theta}{(1 + \sin \theta) + \cos \theta} $$ 21. **Denominator becomes:** $$ ((1 + \cos \theta) - \sin \theta)((1 + \sin \theta) + \cos \theta) $$ 22. **This is a difference of squares form:** $$ (a - b)(a + b) = a^2 - b^2 $$ Where $$a = 1 + \cos \theta$$ and $$b = \sin \theta$$ 23. **Calculate denominator:** $$ (1 + \cos \theta)^2 - \sin^2 \theta = 1 + 2 \cos \theta + \cos^2 \theta - \sin^2 \theta $$ 24. **Use identity $$\cos^2 \theta - \sin^2 \theta = \cos 2\theta$$:** Denominator: $$ 1 + 2 \cos \theta + \cos 2\theta $$ 25. **Calculate numerator:** $$ ((1 + \sin \theta) + \cos \theta)^2 = (1 + \sin \theta)^2 + 2 (1 + \sin \theta) \cos \theta + \cos^2 \theta $$ 26. **Expand numerator:** $$ 1 + 2 \sin \theta + \sin^2 \theta + 2 \cos \theta + 2 \sin \theta \cos \theta + \cos^2 \theta $$ 27. **Use identity $$\sin^2 \theta + \cos^2 \theta = 1$$:** Numerator: $$ 1 + 2 \sin \theta + 1 + 2 \cos \theta + 2 \sin \theta \cos \theta = 2 + 2 \sin \theta + 2 \cos \theta + 2 \sin \theta \cos \theta $$ 28. **Factor numerator:** $$ 2 (1 + \sin \theta + \cos \theta + \sin \theta \cos \theta) $$ 29. **Rewrite denominator using $$\cos 2\theta = 2 \cos^2 \theta - 1$$:** $$ 1 + 2 \cos \theta + 2 \cos^2 \theta - 1 = 2 \cos \theta + 2 \cos^2 \theta $$ 30. **Factor denominator:** $$ 2 \cos \theta (1 + \cos \theta) $$ 31. **Final expression:** $$ \frac{2 (1 + \sin \theta + \cos \theta + \sin \theta \cos \theta)}{2 \cos \theta (1 + \cos \theta)} = \frac{1 + \sin \theta + \cos \theta + \sin \theta \cos \theta}{\cos \theta (1 + \cos \theta)} $$ 32. **Rewrite numerator:** $$ (1 + \sin \theta)(1 + \cos \theta) $$ 33. **Cancel common factor $$1 + \cos \theta$$:** $$ \frac{(1 + \sin \theta)(1 + \cos \theta)}{\cos \theta (1 + \cos \theta)} = \frac{1 + \sin \theta}{\cos \theta} $$ **This matches the right-hand side (RHS), so the identity is proven.**