prove the following identity

sec⁴ A ( 1 - sin ⁴ A ) - 2 tan ² A = 1

Hi Jasmine,

LHS:
= ​sec⁴A (1- sin⁴A) - 2tanA
​=1-sin⁴A_ 2tan²A [since sec A = 1/cos A]
cos⁴A
​= (1 - sin²A) (1 + sin²A) _ 2sin²A [Using (a+b)(a-b) = a2 - b2]
(1 - sin²A) (1 - sin²A) cos²A [cos²A= 1 - sin²A, therefore cos⁴A = (1 - sin²A)²]
​= 1 + sin²A _2sin²A
​ cos²A cos²A
= 1 + sin²A - 2sin²A
cos²A
​= 1 - sin²A
cos²A
= sec²A - tan²A
= 1 [Since tan²A + 1 = sec²A]
= R.H.S
Hence proved.

I hope this helps.
Cheers!