If x=m\sec A+n\tan A and y=m\tan A+n\sec A, then what is x^{2}-y^{2} equal to?

Explanation

Expanding x^2 - y^2, the 2mn\sec A\tan A terms cancel out. We are left with m^2\sec^2 A + n^2\tan^2 A - m^2\tan^2 A - n^2\sec^2 A = m^2(\sec^2 A - \tan^2 A) - n^2(\sec^2 A - \tan^2 A) = m^2(1) - n^2(1) = m^2 - n^2.

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