Question

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

Answer 1

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

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

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

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

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

$=\csc A+\cos A$