Question
Prove.
$\frac{\sin A}{1+\cos A}=\operatorname{cosec} A-\cot A$
$\frac{\sin A}{1+\cos A}=\operatorname{cosec} A-\cot A$
✓
Answer
$\text { LHS }=\frac{\sin A}{1+\cos A} $
$=\frac{\sin A}{1+\cos A} \times \frac{1-\cos A}{1-\cos A} $
$ =\frac{\sin A(1-\cos A)}{1-\cos ^2 A}$
$=\frac{1-\cos A}{\sin A} $
$=\frac{1}{\sin A}-\frac{\cos A}{\sin A}$
$=\operatorname{cosec} A-\cot A=\text { RHS }$
$=\frac{\sin A}{1+\cos A} \times \frac{1-\cos A}{1-\cos A} $
$ =\frac{\sin A(1-\cos A)}{1-\cos ^2 A}$
$=\frac{1-\cos A}{\sin A} $
$=\frac{1}{\sin A}-\frac{\cos A}{\sin A}$
$=\operatorname{cosec} A-\cot A=\text { RHS }$
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