prove that the left side is equal to the right side without using the right side
cot³y - tan³y
= (coty - tany) (cot²y + coty tany + tan²y)
= (cosy/siny - siny/cosy) (cot²y + 1 + tan²y)
= [(cos^2 y - sin^2 y) / (siny cosy] (cot^2 y + sec^2 y)
=> (cot³y - tan³y) / (sec²y + cot²y)
= [(cos^2 y - sin^2 y) / (siny cosy]
= (cos2y) / [(sin2y)/2]
= 2cot(2y)
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cot³y - tan³y
= (coty - tany) (cot²y + coty tany + tan²y)
= (cosy/siny - siny/cosy) (cot²y + 1 + tan²y)
= [(cos^2 y - sin^2 y) / (siny cosy] (cot^2 y + sec^2 y)
=> (cot³y - tan³y) / (sec²y + cot²y)
= [(cos^2 y - sin^2 y) / (siny cosy]
= (cos2y) / [(sin2y)/2]
= 2cot(2y)