Question : If $\sin \theta + \operatorname{cosec} \theta = \sqrt{5}$, the value of $\sin^3 \theta + \operatorname{cosec}^3 \theta = $?
Option 1: $0$
Option 2: $3 \sqrt{5}$
Option 3: $\frac{1}{\sqrt{5}}$
Option 4: $2 \sqrt{5}$
Correct Answer: $2 \sqrt{5}$
Solution :
Given: $\sin \theta + \operatorname{cosec}\ \theta = \sqrt{5}$
Cubing both sides of the given expression, we get,
($\sin\ \theta + \operatorname{cosec} \theta)^3 = (\sqrt{5})^3$
$⇒\sin^3 \theta + \operatorname{cosec}^3 \theta + 3\sin\ \theta × \operatorname{cosec} \theta(\sin\ \theta + \operatorname{cosec} \theta) = (5\sqrt{5})$
$⇒\sin^3 \theta + \operatorname{cosec}^3 \theta + 3\sqrt{5} = 5\sqrt{5}$
$⇒\sin^3 \theta + \operatorname{cosec}^3 \theta = 5\sqrt{5} – 3\sqrt{5}$
$⇒\sin^3 \theta + \operatorname{cosec}^3 \theta = 2\sqrt{5}$
Hence, the correct answer is $2\sqrt{5}$.
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