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Question
Prove the following trigonometric identities
sec4 A(1 − sin4 A) − 2 tan2 A = 1
Solution
We have to prove sec4 A(1 − sin4 A) − 2 tan2 A = 1
We know that `sin^2 A + cos^2 A = 1`
So,
`sec^4 A (1 - sin^4 A) - 2tan^2 A = 1/cos^4 A (1 - sin^4 A) - 2 sin^2 A/cos^2 A`
`= (1/cos^4 A - sin^4 A/cos^4 A) - 2 (sin^2 A)/(cos^2 A)`
`= ((1 - sin^4 A)/cos^4 A) - 2 (sin^2 A)/cos^2 A`
`= ((1 - sin^2 A)(1 + sin^2 A))/cos^4 A - 2 sin^2 A/cos^2 A`
`= (cos^2 A (1 + sin^2 A))/cos^4 A - 2 sin^2 A/cos^2 A`
`= (1 + sin^2 A - 2 sin^2 A)/cos^2 A`
`= (1 - sin^2 A)/cos^2 A`
`= cos^2 A/cos^2 A`
= 1
Hence proved.
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