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Rewrite the problem this way.

1 + sec^4a - tan^4a (I have written sec^4theta first and also for simplicity I am replacing theta by 'a')

=1 + (sec^2a)^2 - (tan^2a)^2 (just splitting the power 4 to squared squared)

= 1 + {(sec^2a + tan^2a)(sec^2a -tan^2a)} using the fact that x^2-y^2 = (x+y)(x-y)

= 1+ {(sec^2a + tan^2a)(1)} this is from a fact(trig identity) that + tan^2a = sec^2a

= 1+ sec^2a + tan^2a

= 1+ tan^2a + sec^2a But 1+ tan^2a = sec^2a

= 2 sec^2a = 2 sec^2theta

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