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Question
Prove the following identities:
sec4 A (1 – sin4 A) – 2 tan2 A = 1
Solution
sec4 A (1 – sin4 A) – 2 tan2 A
= sec4 A – sec4 A sin4 A – 2 tan2 A
= `(sec^2A)^2 - 1/(cos^4A)sin^4A - 2tan^2A`
= (1 + tan2 A)2 – tan4 A – 2 tan2 A ...`[(sec^2A - tan^2A = 1), (sec^2A = 1 + tan^2A)]`
= (1)2 + (tan2 A)2 – 2 × 1 × tan2 A – tan4 A – 2 tan2 A
= 1 + tan4 A + 2 tan2 A – tan4 A – 2 tan2 A
= 1
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