Cosec^6x-cot^6x = 1+3cot^2x+3cot^4x
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cosec^2x - cot^2x=1,
cosec^4x=(1+cot^2x) ^2=1+2cot^2x+cot^4x,
(cosec^2x)^3-(cot^2x)^3
=[(cosec^2x-cot^2x)(cosec^4x+cosec^2x ×cot^2x+cot^4x)],
=[1×{1+2cot^2x+cot^4x+(1+cot^2x) ×cot^2x+cot^4x}]
=[1+3cot^2x+3cot^4x],
so it is proved.
cosec^4x=(1+cot^2x) ^2=1+2cot^2x+cot^4x,
(cosec^2x)^3-(cot^2x)^3
=[(cosec^2x-cot^2x)(cosec^4x+cosec^2x ×cot^2x+cot^4x)],
=[1×{1+2cot^2x+cot^4x+(1+cot^2x) ×cot^2x+cot^4x}]
=[1+3cot^2x+3cot^4x],
so it is proved.
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