Question

Prove that : 2sec^2 A - sec^4 A - 2cosec^2 A + cosec^4 A = cot^4 A - tan^4

Answer 1

LHS:- 2sec^2A- sec^4A -2cosec^2A +cosec^4A

= 2sec^2A - 2cosec^2A +cosec^4A -sec^4A

= 2(1 +tan^2A) - 2(1+cot^2A) +(1 + cot^2A)2 - (1+tan^2A)2

= 2 +2tan^2A - 2 -2cot^2A +1 +cot^4A +2cot^2A - (1 +tan^4A +2tan^2A)

= 2tan^2A - 2cot^2A +1 +cot^4A +2 cot^2A - 1 -tan^4A - 2tan^2A

= cot^4A - tan^4A

= RHS
Hence, proved.