1. **State the problem:** Prove or verify the identity: $$\frac{\cos A - \sin A + 1}{\cos A + \sin A - 1} = \csc A + \cot A$$ using the identity: $$\csc^2 A = 1 + \cot^2 A$$ 2. **Recall important identities:** - $$\csc A = \frac{1}{\sin A}$$ - $$\cot A = \frac{\cos A}{\sin A}$$ - Given: $$\csc^2 A = 1 + \cot^2 A$$ 3. **Start with the left-hand side (LHS):** $$\frac{\cos A - \sin A + 1}{\cos A + \sin A - 1}$$ Try to simplify numerator and denominator by grouping or factoring. 4. **Rewrite numerator and denominator:** Numerator: $$\cos A - \sin A + 1$$ Denominator: $$\cos A + \sin A - 1$$ 5. **Multiply numerator and denominator by the conjugate of the denominator to simplify:** Multiply by $$\frac{\cos A + \sin A + 1}{\cos A + \sin A + 1}$$ 6. **Calculate numerator after multiplication:** $$ (\cos A - \sin A + 1)(\cos A + \sin A + 1) $$ Use distributive property: $$= (\cos A)(\cos A + \sin A + 1) - (\sin A)(\cos A + \sin A + 1) + 1(\cos A + \sin A + 1)$$ $$= \cos^2 A + \cos A \sin A + \cos A - \sin A \cos A - \sin^2 A - \sin A + \cos A + \sin A + 1$$ Simplify terms: $$\cos^2 A - \sin^2 A + \cos A + \cos A + 1$$ Note that $$\cos A \sin A - \sin A \cos A = 0$$ and $$-\sin A + \sin A = 0$$ So numerator becomes: $$\cos^2 A - \sin^2 A + 2 \cos A + 1$$ 7. **Calculate denominator after multiplication:** $$ (\cos A + \sin A - 1)(\cos A + \sin A + 1) $$ This is a difference of squares: $$= (\cos A + \sin A)^2 - 1^2 = (\cos A + \sin A)^2 - 1$$ Expand square: $$= \cos^2 A + 2 \cos A \sin A + \sin^2 A - 1$$ Since $$\cos^2 A + \sin^2 A = 1$$, denominator becomes: $$1 + 2 \cos A \sin A - 1 = 2 \cos A \sin A$$ 8. **Rewrite the fraction:** $$\frac{\cos^2 A - \sin^2 A + 2 \cos A + 1}{2 \cos A \sin A}$$ 9. **Use identity $$\cos^2 A - \sin^2 A = \cos 2A$$:** So numerator is: $$\cos 2A + 2 \cos A + 1$$ 10. **Rewrite numerator as:** $$\cos 2A + 1 + 2 \cos A$$ Recall $$\cos 2A = 2 \cos^2 A - 1$$, so: $$2 \cos^2 A - 1 + 1 + 2 \cos A = 2 \cos^2 A + 2 \cos A$$ 11. **Factor numerator:** $$2 \cos^2 A + 2 \cos A = 2 \cos A (\cos A + 1)$$ 12. **Substitute back into fraction:** $$\frac{2 \cos A (\cos A + 1)}{2 \cos A \sin A}$$ Cancel $$2 \cos A$$ in numerator and denominator: $$\frac{\cancel{2} \cancel{\cos A} (\cos A + 1)}{\cancel{2} \cancel{\cos A} \sin A} = \frac{\cos A + 1}{\sin A}$$ 13. **Rewrite the right-hand side (RHS):** $$\csc A + \cot A = \frac{1}{\sin A} + \frac{\cos A}{\sin A} = \frac{1 + \cos A}{\sin A}$$ 14. **Conclusion:** LHS simplifies to $$\frac{\cos A + 1}{\sin A}$$ which equals RHS. Therefore, the identity is verified. **Final answer:** $$\boxed{\frac{\cos A - \sin A + 1}{\cos A + \sin A - 1} = \csc A + \cot A}$$