Solve
$sec^{6} x - tan^{6} x - 3 sec^{2} x . tan^{2}$

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sec^{6} x - tan^{6} x - 3 sec^{2} x . tan^{2}

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Text solutionVerified

We know that

a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})

Now,

sec^{6} x - tan^{6} x = (sec^{2} x - tan^{2} x) (sec^{4} x + sec^{2} x tan^{2} x + tan^{4} x) = 1. (sec^{4} x + sec^{2} x tan^{2} x + tan^{4} x) = (sec^{2} x - tan^{2} x)^{2} + 2 sec^{2} x tan^{2} x + tan^{2} x sec^{2} x = 1 + 3 sec^{2} x tan^{2} x

Hence,

sec^{6} x - tan^{6} x - 3 sec^{2} x tan^{2} x = 1 + 3 sec^{2} x tan^{2} x - 3 sec^{2} x tan^{2} x = 1

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