Phương trình $\cos 3x.cos^3x - \sin 3xsin^3x = cos^34x + \frac{1}{4}$ tương đương với :
$cos3x.\frac{cos3x+3cosx}{4}-sin3x.\frac{3sinx-sin3x}{4}=cos^34x+\frac{1}{4} $
$\Leftrightarrow \frac{1}{4} coss^23x+\frac{3}{4} cos3xcosx-\frac{3}{4}sin3xsinx+\frac{1}{4} sin^23x=cos^34x+\frac{1}{4} $
$\Leftrightarrow \frac{1}{4} +\frac{3}{4} cos4x=coss^34x+\frac{1}{4} \Leftrightarrow \frac{3}{4} cos4x=cos^34x$
$\Leftrightarrow
4\cos^34x-3\cos 4x=0\Leftrightarrow\cos 12x=0$
$\Leftrightarrow
x=\frac{\pi}{24}+\frac{k\pi}{12}, k\in Z$