Question

Prove that
$\csc ^6a - \cot ^6a= 1 + 3 \csc^2a \cot {a}^{2}$

Answer 1

(Cosec^2A )^3 - (cot^2A)^3 =( cosec^2A - cot^2A )( cosec ^4A + cot^4 A + cosec^2A cot^2A) = 1 + { cosec^4A + cot4A -2 cosec^2A cot ^2 A +2 cosec^2A cot^2A + cosec^2A cot^2A} = 1 + { (cosec^2A - cot^2A ) ^2 + 3 cosec^2A cot ^2A = 1 + 1 + 3 cosec^2Acot^2A = 2 + 3 cosec^2Acot^2A

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