Question

Prove the following trigonometric identities

sec⁴ A(1 − sin⁴ A) − 2 tan² A = 1

Answer 1

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.