Prove that 2sec2θ-sec4θ-2cosec2θ+cosec4θ = cot4θ-tan4θ
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LHS = 2sec2 θ - sec4 θ - 2cosec2 θ + cosec4 θ
Using sec2 θ = tan2 θ + 1 and cosec2 θ = cot2θ + 1, we have
= 2(tan2 θ + 1) - (tan2 θ + 1)2 - 2(cot2 θ + 1) + (cot2 θ + 1)2
= 2tan2 θ + 2 - tan4 θ - 1 -2tan2 θ - 2cot2 θ - 2 + cot4 θ + 1 + 2cot2 θ
= cot4 θ - tan4 θ
= RHS
= Hence proved
Using sec2 θ = tan2 θ + 1 and cosec2 θ = cot2θ + 1, we have
= 2(tan2 θ + 1) - (tan2 θ + 1)2 - 2(cot2 θ + 1) + (cot2 θ + 1)2
= 2tan2 θ + 2 - tan4 θ - 1 -2tan2 θ - 2cot2 θ - 2 + cot4 θ + 1 + 2cot2 θ
= cot4 θ - tan4 θ
= RHS
= Hence proved
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