Question

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Answer 1

QUESTION:-

Prove that Cosec⁶ θ = Cot⁶ θ + 3 Cot² θ Cosec² θ + 1.

GIVEN:-

Cosec⁶ θ = Cot⁶ θ + 3 Cot² θ Cosec² θ + 1.

TO PROVE:-

Cosec⁶ θ = Cot⁶ θ + 3 Cot² θ Cosec² θ + 1.

PROOF:-

Take Cosec⁶ θ as L.H.S.

Take Cot⁶ θ + 3 Cot² θ Cosec² θ + 1 as R.H.S.

⭐L.H.S ⭐

Cosec⁶ θ = (Cosec² θ)³

$\boxed{\bold{ \large{\gray{cosec^2 \ \theta - cot^2 \ \theta= 1}}}}$

$\boxed{\bold{ \large{\gray{cosec^2 \ \theta= 1 + cot^2 \ \theta}}}}$

Cosec⁶ θ = (1 + Cot² θ)³

$\boxed{\bold{ \large{\gray{(A+B)^3=A^3+B^3+3AB(A+B)}}}}$

Cosec⁶ θ = 1³ + (Cot² θ)³ + 3(1)(Cot² θ)(1 + Cot² θ)

Cosec⁶ θ = 1 + Cot⁶ θ + 3Cot² θ(1 + Cot² θ)

$\boxed{\bold{ \large{\gray{cosec^2 \ \theta= 1 + cot^2 \ \theta}}}}$

Cosec⁶ θ = 1 + Cot⁶ θ + 3Cot² θ(Cosec² θ)

Cosec⁶ θ = 1 + Cot⁶ θ + 3 Cot² θ Cosec² θ

Cosec⁶ θ = Cot⁶ θ + 3 Cot² θ Cosec² θ + 1

⭐R.H.S ⭐

Cot⁶ θ + 3 Cot² θ Cosec² θ + 1

L.H.S = R.H.S

Cosec⁶ θ = Cot⁶ θ + 3 Cot² θ Cosec² θ + 1

HENCE PROVED!

VERIFICATION:-

Cosec⁶ θ = Cot⁶ θ + 3 Cot² θ Cosec² θ + 1

Substitute θ = 45°

Cosec⁶ 45° = Cot⁶ 45° + 3 Cot² 45° Cosec² 45° + 1

(√2)⁶ = 1 + 3(1)²(√2)² + 1

(√2)⁶ = (√2 ×√2 × √2 × √2 × √2 × √2)

(√2)⁶ = (2 × 2 × 2)

(√2)⁶ = 8

8 = 1 + 3(2) + 1

8 = 1 + 6 + 1

8 = 8

HENCE VERIFIED!

Answer 2

Answer:

QUESTION:-

Prove that Cosec⁶ θ = Cot⁶ θ + 3 Cot² θ Cosec² θ + 1.

GIVEN:-

Cosec⁶ θ = Cot⁶ θ + 3 Cot² θ Cosec² θ + 1.

TO PROVE:-

Cosec⁶ θ = Cot⁶ θ + 3 Cot² θ Cosec² θ + 1.

PROOF:-

Take Cosec⁶ θ as L.H.S.

Take Cot⁶ θ + 3 Cot² θ Cosec² θ + 1 as R.H.S.

↣L.H.S

Cosec⁶ θ = (Cosec² θ)³

$\boxed{\bold{\pink{cosec^2 \ \theta - cot^2 \ \theta= 1}}}}$

$\boxed{ \large{\blue{cosec^2 \ \theta= 1 + cot^2 \ \theta}}}}$

Cosec⁶ θ = (1 + Cot² θ)³

$\boxed{\bold{ \large{\orange{(A+B)^3=A^3+B^3+3AB(A+B)}}}}$

Cosec⁶ θ = 1³ + (Cot² θ)³ + 3(1)(Cot² θ)(1 + Cot² θ)

Cosec⁶ θ = 1 + Cot⁶ θ + 3Cot² θ(1 + Cot² θ)

$\boxed{\bold{ \large{\green{cosec^2 \ \theta= 1 + cot^2 \ \theta}}}}$

Cosec⁶ θ = 1 + Cot⁶ θ + 3Cot² θ(Cosec² θ)

Cosec⁶ θ = 1 + Cot⁶ θ + 3 Cot² θ Cosec² θ

Cosec⁶ θ = Cot⁶ θ + 3 Cot² θ Cosec² θ + 1

↣R.H.S

Cot⁶ θ + 3 Cot² θ Cosec² θ + 1

L.H.S = R.H.S

Cosec⁶ θ = Cot⁶ θ + 3 Cot² θ Cosec² θ + 1

HENCE PROVED!

VERIFICATION:-

Cosec⁶ θ = Cot⁶ θ + 3 Cot² θ Cosec² θ + 1

Substitute θ = 45°

Cosec⁶ 45° = Cot⁶ 45° + 3 Cot² 45° Cosec² 45° + 1

(√2)⁶ = 1 + 3(1)²(√2)² + 1

(√2)⁶ = (√2 ×√2 × √2 × √2 × √2 × √2)

(√2)⁶ = (2 × 2 × 2)

(√2)⁶ = 8

8 = 1 + 3(2) + 1

8 = 1 + 6 + 1

8 = 8

HENCE VERIFIED!