Question

Use identity ${\csc }^{2}A=1+{\cot }^{2}A$ and prove that $\frac{\cos A-\sin A+1}{\cos A+\sin A-1}=\csc A+\cot A$

Answer 1

L.H.S$=\frac{\cos A-\sin A+1}{\cos A+\sin A-1}$

Divide and multiply by $\sin A$

$\frac{\csc A+\cot A-1}{\csc A-\cot A+1}$ using identity ${\csc }^{2}A-{\cot }^{2}A=1$

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

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

$\frac{[1-(\csc A-\cot A)]\times (\csc A+\cot A)}{\cot A-\csc A+1}$

$=\csc A+\cot A$

$=$R.H.S

hence proved.