Question

Prove that
Cosec⁶ A=Cot⁶A+3Cot²ACosec²A+1​

Answer 1

$\large\bold{\underline{\underline{Question:-}}}$

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

$\large\bold{\underline{\underline{Given:-}}}$

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

$\large\bold{\underline{\underline{To \: Prove:-}}}$

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

$\large\bold{\underline{\underline{Proof:-}}}$

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

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

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

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)

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

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.

$\large\bold{\underline{\underline{Verification:-}}}$

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

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