Answer
$-2 \sin 8x \sin x$
Work Step by Step
We know that $\cos a-\cos b=-2 \sin (\dfrac{a+b}{2}) \sin (\dfrac{a-b}{2})$
$\cos 9 x-\cos 7 x=-2 \sin (\dfrac{9x+7x}{2}) \sin (\dfrac{9x-7x}{2})$
This implies that $\cos 9 x-\cos 7 x=-2 \sin (\dfrac{16x}{2}) \sin (\dfrac{2x}{2})$
Therefore, $\cos 9 x-\cos 7 x=-2 \sin 8x \sin x$
