Question

sin theta +cosec theta=1;coseccube theta +sincube theta=​

Answer 1

Answer:-

Given:-

sin θ + cosec θ = 1 -- equation (1)

Cubing both sides we get,

⟹ (sin θ + cosec θ)³ = (1)³

Using (a + b)³ = a³ + b³ + 3ab(a + b) we get,

⟹ sin³ θ + cosec³ θ + 3 sin θ cosec θ (sin θ + cosec θ) = 1 [ ∵ 1³ = 1 ]

using cosec θ = 1/sin θ and sin θ + cosec θ = 1 [ from (1) ] we get,

⟹ sin³ θ + cosec³ θ + 3 sin θ (1/sin θ) (1) = 1

⟹ sin³ θ + cosec³ θ + 3(1) = 1

⟹ sin³ θ + cosec³ θ = 1 - 3

⟹ sin³ θ + cosec³ θ = - 2

∴ The value of sin³ θ + cosec³ θ is (- 2).

Answer 2

Answer:

Given:-

• sin θ + cosec θ = 1 ____ {1}

To Find:-

• cosec³ θ + sin³ θ

Identity Used:-

• a³ + b³ = (a + b) (a² - ab + b²)

• (a + b)² = a² + b² + 2ab

Solution:-

sin θ + cosec θ = 1

Squaring both side,

(sin θ + cosec θ)² = 1²

Using identity (a + b)² = a² + b² + 2ab

⇒ sin² θ + cosec² θ + 2sinθ cosecθ = 1

⇒ sin² θ + cosec² θ = 1 - 2

⇒ sin² θ + cosec² θ = -1 ____ {2}

Now, using identity,

a³ + b³ = (a + b) (a² - ab + b²) in {1}

⇒ sin³ θ + cosec³ θ = (sin θ + cosec θ) (sin² θ - sin θ cosec θ + cosec² θ)

⇒ sin³ θ + cosec³ θ = 1 (sin² θ + cosec² θ -1 ) {from 1}

⇒ sin³ θ + cosec³ θ = 1 × (-1 - 1) {from 2}

⇒ sin³ θ + cosec³ θ = 1 × (-2)

⇒ sin³ θ + cosec³ θ = -2

Hence, the value of sin³ θ + cosec³ θ is -2.

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