If sin θ + cosec \ θ = √(5) , then the value of sin³ θ + cosec³ θ = __.

If sin θ + cosec \ θ = √(5) , then the value of sin³ θ +…

If $\sin \theta + \text{cosec} \ \theta = \sqrt{5}$ , then the value of sin³ θ + cosec³ θ = ______.

Show answer & explanation

Correct answer: D

Explanation

The correct answer is option 4 i.e. $2\sqrt{5}$ .

sin θ + cosec θ = √5

⇒ (sin θ + cosec θ)² = (√5)²

⇒ sin²θ + 2 sin θ cosec θ + cosec²θ = 5

⇒ sin²θ + cosec²θ = 5 - 2 = 3

Now, sin³θ + cosec³θ = (sin θ + cosec θ)(sin²θ - sin θ cosec θ + cosec²θ)

= (√5)(sin²θ - 1 + cosec²θ)

= (√5)(sin²θ + cosec²θ - 1)

= (√5)(3 - 1)

= 2√5

Therefore, the required value is 2√5.