Question

Prove that :
cosec^6 A - cot^6 A=3cot^2 A . cosec^2 A+1

Answer 1

Step-by-step explanation:

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Answer 2

It is proven that cosec⁶A – cot⁶A = 3cosec² A. cot² A = 1  

Given:

cosec⁶ A - cot⁶ A = 3cot² A.cosec² A+1

To find:

Prove that cosec⁶ A - cot⁶ A = 3cot² A.cosec² A+1

Solution:

As we from trigonometric identities cosec² θ - cot² θ = 1  

=> cosec² A - cot² A = 1    

On taking the cube on both sides

=> (cosec² A - cot² A)³ = 1³    

From (a – b)³ = a³ – b³ – 3ab(a – b)  

=> (cosec² A)³ – (cot² A)³ – 3(cosec² A)(cot² A)(cosec² A - cot² A) = 1

=> (cosec⁶A) – cot⁶A – 3cosec² A. cot² A (1) = 1  

=> cosec⁶A – cot⁶A – 3cosec² A. cot² A = 1  

=> cosec⁶A – cot⁶A = 3cosec² A. cot² A = 1  

Hence,

It is proven that cosec⁶A – cot⁶A = 3cosec² A. cot² A = 1  

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