Question

$\frac{\cos A-\sin A+1}{\cos A+\sin A-1}=\csc A+\cot A,u\sin g$ the identity ${\csc }^{2}A=1+{\cot }^{2}A$

Answer 1

given, $\frac{\cos A-\sin A+1}{\cos A+\sin A-1}$

$=\frac{\cot A-1+cosecA}{\cot A+1-cosecA}$

(Dividing Numerator and Denominator by $\sin A$)

$=\frac{(\cot A+cosecA)-(cose{c}^{2}A-{\cot }^{2}A)}{1+\cot A-cosecA}$

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

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

$=\cot A+cosecA$

Hence proved.