Question
Prove.$\frac{\cot ^2 A}{(\operatorname{cosec} A+1)^2}=\frac{1-\sin A}{1+\sin A}$
✓
Answer
$\text { R.H.S }=\frac{1-\sin A}{1+\sin A}$
$=\frac{1-\frac{1}{\operatorname{cosec} A}}{1+\frac{1}{\operatorname{cosec} A}}=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}$
$=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1} \times \frac{\operatorname{cosec}+1}{\operatorname{cosec} A+1}$
$=\frac{\operatorname{cosec} A-1}{(\operatorname{cosec} A+1)^2}=\frac{\cot ^2 A}{(\operatorname{cosec} A+1)^2}\left(\because \operatorname{cosec}^2 A-1=\cot ^2 A\right)$
$=\text { L.H.S }$
$=\frac{1-\frac{1}{\operatorname{cosec} A}}{1+\frac{1}{\operatorname{cosec} A}}=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}$
$=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1} \times \frac{\operatorname{cosec}+1}{\operatorname{cosec} A+1}$
$=\frac{\operatorname{cosec} A-1}{(\operatorname{cosec} A+1)^2}=\frac{\cot ^2 A}{(\operatorname{cosec} A+1)^2}\left(\because \operatorname{cosec}^2 A-1=\cot ^2 A\right)$
$=\text { L.H.S }$
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