Evaluate $\displaystyle \lim_{x\rightarrow 0} \frac{1-\cos4x}{x^2}$

so by the double angle

$\displaystyle \cos 4x = \cos^2 2x - \sin^2 2x $

this I understand but it shows it go further to $\displaystyle 1-2\sin^2 2x$

This is where it starts to lose me.