# How do you prove cos^4(x)-sin^4(x)=cos2x?

Apr 19, 2015

${\cos}^{4} \left(x\right) - {\sin}^{4} \left(x\right) = \cos \left(2 x\right)$

Remember the double angle formula for cosine:
color(blue)(cos(2x)=cos^2(x)-sin^2(x)

Plugging it into the right hand side:
${\cos}^{4} \left(x\right) - {\sin}^{4} \left(x\right) = {\cos}^{2} \left(x\right) - {\sin}^{2} \left(x\right)$

Using differences of squares on the left side:
$\left({\cos}^{2} \left(x\right) + {\sin}^{2} \left(x\right)\right) \left({\cos}^{2} \left(x\right) - {\sin}^{2} \left(x\right)\right) = {\cos}^{2} \left(x\right) - {\sin}^{2} \left(x\right)$

And since color(blue)(cos^2(x) + sin^2(x)=1
$1 \left({\cos}^{2} \left(x\right) - {\sin}^{2} \left(x\right)\right) = {\cos}^{2} \left(x\right) - {\sin}^{2} \left(x\right)$