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