Question : Simplify the given equation: $\frac{\cot^3A–1}{\cot A–1}$
Option 1: $\operatorname{cosec}^2 \mathrm{A}-\cot \mathrm{A}$
Option 2: $\operatorname{cosec}^2 A+\cot A$
Option 3: $\cot ^2 \mathrm{A}+\operatorname{cosec} \mathrm{A}$
Option 4: $\cot ^2 \mathrm{A}-\operatorname{cosec} \mathrm{A}$
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Correct Answer: $\operatorname{cosec}^2 A+\cot A$
Solution :
Given:
$\frac{\cot^3A–1}{\cot A–1}$
= $\frac{(\cot A–1)(\cot^2A+\cot A+1)}{(\cot A–1)}$
= $\cot^2A+\cot A+1$
= $\operatorname{cosec^2}A+\cot A$
Hence, the correct answer is $\operatorname{cosec}^2 A+\cot A$.
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