$A=\dfrac{\sin^4x-\cos^4x+\sin2x}{\tan2x-1}$
$=\dfrac{\sin^2x-\cos^2x+\sin2x}{\tan2x-1}$
$=\dfrac{\sin^2x-\cos^2x+\sin2x}{\frac{\sin2x-\cos2x}{\cos2x}}$
$=\dfrac{\cos2x(\sin^2x-\cos^2x+\sin2x)}{\sin2x-\cos2x}$
$=\dfrac{\cos2x(-\cos 2x+\sin2x)}{-\cos2x+\sin2x}$
$=\cos 2x$