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How do you prove ##cos2x=cos^2x-sin^2## using other trigonometric identities?

    Apply the angle-sum identity for cosine to ##cos(x+x)##.

    The identity needed is the angle-sum identity for cosine.

    ##cos(alpha + beta) = cos(alpha)cos(beta) – sin(alpha)sin(beta)##

    With that, we have

    ##cos(2x) = cos(x + x)##

    ##= cos(x)cos(x) – sin(x)sin(x)##

    ##= cos^2(x) – sin^2(x)##

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