[tex]\displaystyle\\
\frac{\cos x + \cos 5x}{\cos 3x + \cos x} - \frac{\cos 5x - \cos x}{\cos 3x - \cos x} =\\\\
=\frac{\cos 5x + \cos x}{\cos 3x + \cos x} - \frac{\cos 5x - \cos x}{\cos 3x - \cos x} =\\\\
=\frac{2\cos\dfrac{5x+x}{2}\cos\dfrac{5x-x}{2}}{2\cos\dfrac{3x+x}{2}\cos\dfrac{3x-x}{2}} - \frac{-2\sin\dfrac{5x+x}{2}\sin\dfrac{5x-x}{2}}{-2\sin\dfrac{3x+x}{2}\sin\dfrac{3x-x}{2}} =[/tex]
[tex]\displaystyle\\
=\frac{\cos 3x\cos 2x }{\cos 2x \cos x } - \frac{\sin 3x \sin 2x}{\sin 2x \sin x
} =\\\\
=\frac{\cos 3x}{\cos x}-\frac{\sin 3x}{\sin x} =\\\\
=\frac{4\cos^3 x -3\cos x}{\cos x}-\frac{3\sin x-4\sin^3 x}{\sin x} =\\\\
=\frac{\cos x(4\cos^2 x -3)}{\cos x}-\frac{\sin x(3-4\sin^2 x)}{\sin x} =\\\\
=(4\cos^2 x -3) - (3-4\sin^2 x)=\\\\
=4\cos^2 x -3 - 3+4\sin^2 x=\\\\
=4(\sin^2 x + \cos^2 x)-6 = 4\cdot 1-6 = 4-6 = \boxed{\bf -2}[/tex]
