Question

$\frac{\cos A-\sin A+1}{\cos A+\sin A-1}=\cos ecA+\cot A,$ using the identity $\cos e{c}^{2}A=1+{\cot }^{2}A.$

Answer 1

$\frac{cosA-sinA+1}{cosA+sinA-1}=\frac{cotA-1+cosecA}{cotA+1-cosecA}$ (Divide numerator and denominator by $sinA$ )

$=\frac{cotA-cose{c}^{2}A+co{t}^{2}A+cosecA}{cotA+1-cosecA}$ (Using $cose{c}^{2}A-co{t}^{2}A=1$)

$=\frac{(cotA+cosecA)+(cotA+cosecA)(cotA-cosecA)}{cotA+1-cosecA}=\frac{(cotA+cosecA)(1+cotA-cosecA)}{cotA+1-cosecA}=cotA+cosecA$