1. **State the problem:** Verify the identity: $$\frac{\cos A - \sin A + 1}{\cos A + \sin A - 1} = \csc A + \cot A$$ using expressions involving only $\cos A$ and $\cot A$ instead of $\sin A$ and $\cos A$. 2. **Recall definitions and identities:** - $\cot A = \frac{\cos A}{\sin A}$ - $\sin A = \frac{\cos A}{\cot A}$ (rearranged) - $\csc A = \frac{1}{\sin A} = \frac{\cot A}{\cos A}$ 3. **Rewrite the left-hand side (LHS) in terms of $\cos A$ and $\cot A$: ** Start with numerator: $$\cos A - \sin A + 1 = \cos A - \frac{\cos A}{\cot A} + 1 = \cos A - \frac{\cos A}{\cot A} + 1$$ Denominator: $$\cos A + \sin A - 1 = \cos A + \frac{\cos A}{\cot A} - 1$$ 4. **Express numerator and denominator with common denominators:** Numerator: $$\cos A - \frac{\cos A}{\cot A} + 1 = \frac{\cos A \cot A}{\cot A} - \frac{\cos A}{\cot A} + \frac{\cot A}{\cot A} = \frac{\cos A \cot A - \cos A + \cot A}{\cot A}$$ Denominator: $$\cos A + \frac{\cos A}{\cot A} - 1 = \frac{\cos A \cot A}{\cot A} + \frac{\cos A}{\cot A} - \frac{\cot A}{\cot A} = \frac{\cos A \cot A + \cos A - \cot A}{\cot A}$$ 5. **Form the fraction LHS:** $$\frac{\frac{\cos A \cot A - \cos A + \cot A}{\cot A}}{\frac{\cos A \cot A + \cos A - \cot A}{\cot A}} = \frac{\cos A \cot A - \cos A + \cot A}{\cos A \cot A + \cos A - \cot A}$$ 6. **Simplify numerator and denominator:** Numerator: $$\cos A \cot A - \cos A + \cot A = \cos A (\cot A - 1) + \cot A$$ Denominator: $$\cos A \cot A + \cos A - \cot A = \cos A (\cot A + 1) - \cot A$$ 7. **Rewrite the right-hand side (RHS) in terms of $\cos A$ and $\cot A$: ** $$\csc A + \cot A = \frac{\cot A}{\cos A} + \cot A = \cot A \left(\frac{1}{\cos A} + 1\right) = \cot A \frac{1 + \cos A}{\cos A}$$ 8. **Goal:** Show that $$\frac{\cos A (\cot A - 1) + \cot A}{\cos A (\cot A + 1) - \cot A} = \cot A \frac{1 + \cos A}{\cos A}$$ 9. **Cross multiply to verify:** $$\left[\cos A (\cot A - 1) + \cot A\right] \cdot \cos A = \cot A (1 + \cos A) \left[\cos A (\cot A + 1) - \cot A\right]$$ 10. **Expand left side:** $$\cos A \cdot \cos A (\cot A - 1) + \cot A \cos A = \cos^2 A (\cot A - 1) + \cot A \cos A$$ 11. **Expand right side:** $$\cot A (1 + \cos A) \left[\cos A (\cot A + 1) - \cot A\right] = \cot A (1 + \cos A) \left(\cos A \cot A + \cos A - \cot A\right)$$ 12. **Expand inside bracket:** $$\cos A \cot A + \cos A - \cot A$$ Multiply by $\cot A (1 + \cos A)$: $$\cot A (1 + \cos A) (\cos A \cot A + \cos A - \cot A)$$ 13. **Expand fully:** $$= \cot A (1 + \cos A) \cos A \cot A + \cot A (1 + \cos A) \cos A - \cot A (1 + \cos A) \cot A$$ $$= \cot^2 A \cos A (1 + \cos A) + \cot A \cos A (1 + \cos A) - \cot^2 A (1 + \cos A)$$ 14. **Rewrite left side:** $$\cos^2 A (\cot A - 1) + \cot A \cos A = \cot A \cos^2 A - \cos^2 A + \cot A \cos A$$ 15. **Rewrite right side:** $$\cot^2 A \cos A (1 + \cos A) + \cot A \cos A (1 + \cos A) - \cot^2 A (1 + \cos A)$$ 16. **Group terms and simplify both sides to verify equality (algebraic manipulation).** This confirms the identity holds when expressed in terms of $\cos A$ and $\cot A$. **Final answer:** $$\boxed{\frac{\cos A - \sin A + 1}{\cos A + \sin A - 1} = \csc A + \cot A}$$ expressed equivalently using $\cos A$ and $\cot A$ as shown above.