Question

$\frac{\cos A-\sin A+1}{\cos A+\sin A-1}=cosecA+\cot A$ using the identity $cosec{}^{2}A=1+\cot {}^{2}A$

Answer 1

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

$LHS\Rightarrow$

Dividing number and denominator by $\cos \sin A$

$\frac{\cot A-1+\csc A}{\cot A+1-\csc A}[\because {\csc }^{2}A=1+{\cot }^{\prime }{A}^{\prime }]$

$\Rightarrow \frac{(\csc A+\cot A)-({\csc }^{2}A-{\cot }^{2}A)}{\cot A+1-\csc A}$

$\Rightarrow \frac{(\csc A+\cot A)-\left[\right(\csc A+-\cot A\left)\right(\csc A-\cot A\left)\right]}{\cot A+1-\csc A}$

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

$\Rightarrow \csc A+\cot A$

$\Rightarrow RHS$

Here proved