Question : Given $A$ is an acute angle, what is the value of $\left(1-\sin ^2 A\right) \operatorname{cosec}^2 A$?
Option 1: $\cot ^2 \mathrm{~A}$
Option 2: $\cos ^2 \mathrm{~A}$
Option 3: $\tan ^2 \mathrm{~A}$
Option 4: $\sin ^2 \mathrm{~A}$
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Correct Answer: $\cot ^2 \mathrm{~A}$
Solution :
Given: $(1-\sin ^2 A) \operatorname{cosec}^2 A$
We know, $\sin^2A + \cos^2A=1$ and $\operatorname{cosec} A=\frac{1}{\sin A}$
So, $(1-\sin ^2 A) \operatorname{cosec}^2 A$
$=\cos^2A\cdot\frac{1}{\sin^2A}$
$=\cot^2A$
Hence, the correct answer is $\cot^2A$.
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