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LHS = 2 sec²A - sec⁴A - 2cosec²A + cosec⁴A
= 2 sec²A - 2cosec²A - sec⁴A + cosec⁴A
= 2(sec²A - cosec²A) -[(sec²A)² - (cosec²A)²] =2(sec²A-cosec²A)+[(sec²A-cosec²A)(sec²A+cosec²A)]
= (sec²A - cosec²A) [2 - (sec²A + cosec²A)]
= (sec²A - cosec²A) (2 - sec²A - cosec²A)
= (sec²A - cosec²A) (1 - sec²A + 1 - cosec²A)
= (sec²A - cosec²A) (tan²A - cot²A)
= (1 - tan²A - 1 - cot²A) (tan²A - cot²A)
= - (tan²A + cot²A) (tan²A - cot²A)
= - (tan⁴A + cot⁴A)
= cot⁴A - tan⁴A
= RHS
.Proved
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