Question

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

Answer 1

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

$=\frac{1}{\frac{1}{\sin A}-\frac{\cos A}{\sin A}}$

$=\frac{\sin A}{1-\cos A}$

$\frac{\sin A}{1-\cos A}=\frac{\sin A(1+\cos A)}{1-{\cos }^{2}A}$

$=\frac{\sin A(1+\cos A)}{{\sin }^{2}A}$

$=\frac{1+\cos A}{\sin A}$

$\frac{\sin A}{1-\cos A}=\frac{1+\cos A}{\sin A}=\frac{\cos A-\sin A+1}{\cos A+\sin A-1}$

$[\because If\frac{x}{a}=\frac{y}{b}(ab\ne 0)then\frac{x}{a}=\frac{y}{b}=\frac{x-y}{a-b}(a\ne b\left)\right]$