1. **State the problem:** Prove that $$\cos A - \sin A + 1 = \csc A + \cot A$$ using the identity $$\csc^2 A = 1 + \cot^2 A$$. 2. **Recall the identity:** We know that $$\csc^2 A = 1 + \cot^2 A$$. 3. **Express the right side in terms of sine and cosine:** $$\csc A + \cot A = \frac{1}{\sin A} + \frac{\cos A}{\sin A} = \frac{1 + \cos A}{\sin A}$$. 4. **Rewrite the left side:** $$\cos A - \sin A + 1$$. 5. **Goal:** Show that $$\cos A - \sin A + 1 = \frac{1 + \cos A}{\sin A}$$. 6. **Multiply both sides of the equation by $$\sin A$$ to clear the denominator:** $$\sin A (\cos A - \sin A + 1) = 1 + \cos A$$. 7. **Expand the left side:** $$\sin A \cos A - \sin^2 A + \sin A = 1 + \cos A$$. 8. **Rearrange terms:** $$\sin A \cos A + \sin A - \sin^2 A = 1 + \cos A$$. 9. **Use the Pythagorean identity:** $$\sin^2 A = 1 - \cos^2 A$$, so $$-\sin^2 A = -(1 - \cos^2 A) = -1 + \cos^2 A$$. 10. **Substitute:** $$\sin A \cos A + \sin A - 1 + \cos^2 A = 1 + \cos A$$. 11. **Group terms:** $$\sin A \cos A + \sin A + \cos^2 A - 1 = 1 + \cos A$$. 12. **Bring all terms to one side:** $$\sin A \cos A + \sin A + \cos^2 A - 1 - 1 - \cos A = 0$$ $$\sin A \cos A + \sin A + \cos^2 A - \cos A - 2 = 0$$. 13. **This expression is not identically zero, so the original equality does not hold for all $$A$$.** 14. **Conclusion:** The given equation $$\cos A - \sin A + 1 = \csc A + \cot A$$ is not true in general, so it cannot be proven using the given identity. **Final answer:** The equation is false and cannot be proven as stated.