Prove the following.
sec6x – tan6x = 1 + 3 sec2x × tan2x
Taking LHS
sec6x - tan6x
= (sec2x)3 - (tan2x)3
= (sec2x - tan2x)(sec4x + tan2x sec2x + tan4x)
[As, a3 - b3 = (a - b)(a2 + ab + b2)]
= sec4x + tan4x + tan2x sec2x + 2tan2x sec2x - 2tan2x sec2x
[As, sec2θ - tan2θ = 1]
= sec4x + tan4x - 2tan2x sec2x + 3tan2x sec2x
= (sec2x - tan2x)2 + 3tan2x sec2x [a2 + b2 - 2ab = (a - b)2]
= 12 + 3tan2x sec2x
= 1 + 3tan2x sec2x
= RHS
Proved.
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