Question

Prove that:
cosec^6A-cot^6A=3cot²Acosec²A+1​

Answer 1

$\bf\large\underline\pink{Answer:-}$

Answer 2

Answer:

LHS

= cosec^6 A - cot^6 A

= (cosec^2 A)^3 - (cot^2 A)^3

= (cosec^2 A - cot^2 A)(cosec^4 A + cot^4 A + cosec^2 A * cot^2 A)

= (cosec^4 A + cot^4 A + cosec^2 A * cot^2 A

Since, cosec^2 A - cot^2 A = 1

= (cot^2 A + 1)^2 + cot^4 A + (cot^2 A +1)cot^2 A

= cot^4 A + 2cot^2 A + 1 + cot^4 A + cot^4 A + cot^2 A

= 3cot^4 A + 3cot^2 A + 1

Now,

RHS = 3cot^2 Acosec^2 A + 1

= 3(cot^2 A) (cot^2 A + 1) + 1

= 3 cot^4 A + 3 cot ^2 A + 1

Clearly, LHS = RHS

Hence proved the identity that

cosec^6 A - cot^6 A = 3cot^2 A cosec^2 A + 1

Hope this helps you !