# Prove that \sin^{2}x-\cos^{2}=1-2cos^{2}x

Prove that $$\displaystyle{{\sin}^{{{2}}}{x}}-{{\cos}^{{{2}}}=}{1}-{2}{{\cos}^{{{2}}}{x}}$$

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Mary Moen
Look, we have:
$$\displaystyle{{\sin}^{{{2}}}-}{{\cos}^{{{2}}}{x}}={{\sin}^{{{2}}}{x}}+{\left({{\cos}^{{{2}}}{x}}-{{\cos}^{{{2}}}{x}}\right)}-{{\cos}^{{{2}}}{x}}={\left({{\sin}^{{{2}}}{x}}+{\cos}^{{{2}}}\right)}-{2}{{\cos}^{{{2}}}{x}}={1}-{2}{{\cos}^{{{2}}}{x}}$$
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Ceitheart

Just start with $$\displaystyle{{\sin}^{{{2}}}{x}}+{{\cos}^{{{2}}}{x}}={1}$$ and subtract
$$2\cos^{2}x$$ from both sides. And here you go!