How do you prove that cos^4x-sin^4x=cos(2x) ?

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How do you prove that cos^4x-sin^4x=cos(2x) ?

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Answer 1 · 2018-05-22T14:13:25

$f (x) = {cos}^{4} x - {sin}^{4} x = ({cos}^{2} x - {sin}^{2} x) ({cos}^{2} x + {sin}^{2} x)$ $f(x) = cos^4 x - sin^4 x = (cos^2 x - sin^2 x)(cos^2 x + sin^2 x)$
Reminder of trig identities:
${cos}^{2} x - {sin}^{2} x = cos 2 x$ $cos^2 x - sin^2 x = cos 2x$
${cos}^{2} x + {sin}^{2} x = 1$ $cos^2 x + sin^2 x = 1$
Therefor,
f(x) = cos 2x. Proved.

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How do you prove that cos^4x-sin^4x=cos(2x) ?

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