Question

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

Answer 1

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

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

$=\frac{[\cos A-(\sin A-1\left)\right]\cos A-(\sin A-1)}{\cos A+(\sin A-1)\cos A-(\sin A-1)}$

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

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

$=\frac{2-2\cos A\cdot \sin A+2\cos A-2\sin A}{1-2{\sin }^{2}A-1+2\sin A}$ $[\because {\cos }^{2}A+{\sin }^{2}A=1]$

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

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

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

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