How Many Solutions Are There?

$\cos(6x\pm x) \equiv \cos 6x \cdot \cos x \mp \sin 6x \cdot \sin x$

The equation we have to solve is

$\cos(6x + x)=\cos(6x - x)$

By the above identity, this is just $\sin 6x \cdot \sin x = 0$ , which has the $\boxed7$ solutions

$0,\frac{\pi }{6},\frac{\pi }{3},\frac{\pi }{2},\frac{2\pi }{3},\frac{5\pi }{6},\pi$

in the interval $[0,\pi]$

$\begin{aligned} \cos (7x) & = \cos (5x) \\ \cos (5x) - \cos (7x) & = 0 \\ \cos (6x-x) - \cos (6x+x) & = 0 \\ 2 \sin (6x) \sin x & = 0 \\ \implies x & = 0, \frac \pi 6, \frac \pi 3, \frac \pi 2, \frac {2\pi}3, \frac {5\pi}6, \pi & \small \text{for }x \in [0. \pi]\end{aligned}$

Therefore, there are $\boxed 7$ solutions.