Question : If $0^{\circ}< \theta< 90^{\circ}$ and $\operatorname{cosec \theta} =\cot^{2}\theta$, then the value of expression $\operatorname{cosec^{4}\theta}–\operatorname{2cosec^{2}\theta}-\cot^{2}\theta$ is equal to:
Option 1: $2$
Option 2: $0$
Option 3: $1$
Option 4: $3$
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Correct Answer: $0$
Solution :
$\operatorname{cosec \theta} =\cot^{2}\theta$
$⇒\operatorname{cosec \theta} = \operatorname{cosec^2 \theta} -1$
Squaring both sides, we get,
$⇒\operatorname{cosec^2 \theta} =\operatorname{cosec^4 \theta} -2\operatorname{cosec^2 \theta} +1$
$⇒\operatorname{cosec^4 \theta} -3\operatorname{cosec^2 \theta} +1=0$
$⇒\operatorname{cosec^4 \theta} -3\operatorname{cosec^2 \theta} +\operatorname{cosec^2 \theta} -\cot^2\theta =0$ [$\because1 = \operatorname{cosec^2 \theta}-\cot^2\theta$]
$⇒\operatorname{cosec^4 \theta} -2\operatorname{cosec^2 \theta} -\cot^2\theta =0$
Hence, the correct answer is $0$.
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