Question : If $\theta$ is an acute angle and $\sin \theta+\operatorname{cosec} \theta=2$, then the value of $\sin ^5 \theta+\operatorname{cosec}^5 \theta$ is:
Option 1: 10
Option 2: 2
Option 3: 4
Option 4: 5
Correct Answer: 2
Solution :
Given,
$\sin \theta+\operatorname{cosec} \theta=2$
This is possible only when both $\sin \theta=\operatorname{cosec} \theta=1$
So, $\sin ^5 \theta+\operatorname{cosec}^5 \theta=1^5+1^5$
⇒ $\sin \theta+\operatorname{cosec} \theta=1+1$
⇒ $\sin \theta+\operatorname{cosec} \theta=2$
Hence, the correct answer is 2.
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