Question
Prove that$\frac{\cot ^2 A}{\operatorname{cosec} A-1}-1=\operatorname{cosec} A$
✓
Answer
$\frac{\cot ^2 A}{\operatorname{cosec} A-1}-1$
$=\frac{\cot ^2 A-\operatorname{cosec} A+1}{\operatorname{cosec} A-1}$
$=\frac{-\operatorname{cosec} A+\operatorname{cosec}{ }^2 A}{\operatorname{cosec} A-1}$
$=\frac{\operatorname{cosec} A(\operatorname{cosec} A-1)}{\operatorname{cosec} A-1}$
$=\operatorname{cosec} A$
$=\frac{\cot ^2 A-\operatorname{cosec} A+1}{\operatorname{cosec} A-1}$
$=\frac{-\operatorname{cosec} A+\operatorname{cosec}{ }^2 A}{\operatorname{cosec} A-1}$
$=\frac{\operatorname{cosec} A(\operatorname{cosec} A-1)}{\operatorname{cosec} A-1}$
$=\operatorname{cosec} A$
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