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math

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prove the identity :
cos^4 theta- sin ^4 thetaover sin squared theta cos squared theta =cot squared theta - tan squared theta

  • math -

    I'll use x for theta, for my convenience.

    (cos^4x - sin^4x)/sin^2x cos^2x

    = (cos^2x - sin^2x)(cos^2x + sin^2x)/sin^2x cos^2x

    = (cos^2x - sin^2x)/sin^2x * (cos^2x + sin^2x)/cos^2x

    = (cot^2x - 1)(1+tan^2x)
    = cot^2x - 1 + cot^2xtan^2x - tan^2x
    = cot^2x - 1 + 1 - tan^2x
    = cot^2x - tan^2x

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