Question

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

Answer 1

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

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

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

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

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

$=\frac{(\sin A+1)}{\cos A}=$R.H.S