1. **State the problem:** Verify or simplify the expression $$\frac{1+\cos A + \sin A}{1 - \cos A + \sin A} = \cot \frac{A}{2}$$. 2. **Recall the half-angle formulas:** $$\cot \frac{A}{2} = \frac{1 + \cos A}{\sin A}$$ 3. **Rewrite the left-hand side (LHS):** $$\frac{1 + \cos A + \sin A}{1 - \cos A + \sin A}$$ 4. **Try to simplify the LHS by multiplying numerator and denominator by the conjugate of the denominator:** Multiply numerator and denominator by $$1 - \cos A - \sin A$$: $$\frac{(1 + \cos A + \sin A)(1 - \cos A - \sin A)}{(1 - \cos A + \sin A)(1 - \cos A - \sin A)}$$ 5. **Calculate denominator:** $$(1 - \cos A)^2 - (\sin A)^2 = 1 - 2\cos A + \cos^2 A - \sin^2 A$$ Using $$\cos^2 A - \sin^2 A = \cos 2A$$: $$= 1 - 2\cos A + \cos 2A$$ 6. **Calculate numerator:** Expand: $$(1)(1 - \cos A - \sin A) + \cos A (1 - \cos A - \sin A) + \sin A (1 - \cos A - \sin A)$$ Simplify stepwise: $$1 - \cos A - \sin A + \cos A - \cos^2 A - \cos A \sin A + \sin A - \sin A \cos A - \sin^2 A$$ Combine like terms: $$1 - \cos^2 A - \sin^2 A - 2 \cos A \sin A$$ Recall $$\cos^2 A + \sin^2 A = 1$$, so numerator becomes: $$1 - 1 - 2 \cos A \sin A = -2 \cos A \sin A$$ 7. **Rewrite denominator using double angle formula:** $$1 - 2 \cos A + \cos 2A = 1 - 2 \cos A + (2 \cos^2 A - 1) = 2 \cos^2 A - 2 \cos A$$ Factor: $$2 \cos A (\cos A - 1)$$ 8. **So the expression is:** $$\frac{-2 \cos A \sin A}{2 \cos A (\cos A - 1)} = \frac{- \sin A}{\cos A - 1}$$ 9. **Rewrite denominator:** $$\cos A - 1 = -(1 - \cos A)$$ So expression becomes: $$\frac{- \sin A}{\cos A - 1} = \frac{- \sin A}{-(1 - \cos A)} = \frac{\sin A}{1 - \cos A}$$ 10. **Recall the half-angle identity:** $$\cot \frac{A}{2} = \frac{1 + \cos A}{\sin A}$$ But we have $$\frac{\sin A}{1 - \cos A}$$, which is the reciprocal of $$\tan \frac{A}{2} = \frac{\sin A}{1 + \cos A}$$. 11. **Check if the original equality holds:** The original expression simplifies to $$\frac{\sin A}{1 - \cos A}$$, which equals $$\tan \frac{A}{2}$$, not $$\cot \frac{A}{2}$$. **Final conclusion:** $$\frac{1 + \cos A + \sin A}{1 - \cos A + \sin A} = \tan \frac{A}{2} \neq \cot \frac{A}{2}$$ Hence, the given equation is incorrect as stated.