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How do you prove ## cos ^2theta – sin ^2 theta = 1 – 2sin^2 theta ##?

    The Pythagorean identity states that ##sin^2theta+cos^2theta=1##.

    We can rearrange the identity to see that ##cos^2theta=1-sin^2theta##.

    If we know that ##cos^2theta=1-sin^2theta##, we can replace ##cos^2theta## with ##1-sin^2theta## in the expression ##cos^2theta-sin^2theta##.

    ##cos^2theta-sin^2theta=(1color(blue)(-sin^2theta))color(blue)(-sin^2theta##

    Combine like terms to see that:

    ##cos^2theta-sin^2theta=1-2sin^2theta##