2. The general solution of the equation $\cos \frac{x}{4}-\cos x$ is ( ), within $(0,24 \pi)$, there are ( ) distinct solutions.
2. The original equation $\cos \frac{x}{4}=\cos x$ can be transformed into
$$
\cos x-\cos \frac{x}{4}=0
$$

which is
$$
-2 \sin \frac{5 x}{8} \sin \frac{3 x}{8}=0 .
$$

From $\sin \frac{5 x}{8}=0$, we get $\frac{5 x}{8}=m \pi, \quad x=\frac{8 m \pi}{5}(m \in \mathbb{Z})$
From $\sin \frac{3 x}{8}=0$, we get $\frac{3 x}{8}=n \pi, \quad x=\frac{8 n \pi}{3}(n \in \mathbb{Z})$.
Notice that the function $f(x)=\cos x-\cos \frac{x}{4}$ is a periodic function with a period of $8 \pi$, and within one period $(0,8 \pi)$, the roots of $f(x)=0$ are
$$
0, \frac{8 \pi}{5}, \frac{16 \pi}{5}, \frac{24 \pi}{5}, \frac{32 \pi}{5} \text { and } \frac{8 \pi}{3}, \frac{16 \pi}{3}
$$

There are 7 roots in total, so the number of distinct solutions in $(0,24 \pi)$ is
$$
3 \times 7-1=20 \text { solutions. }
$$
$\therefore$ The general solution of $\cos \frac{x}{4}=\cos x$ is
$$
x=\frac{8 m}{5} \pi \quad \text { or } \quad x=\frac{8 n \pi}{3},
$$

where $m, n \in \mathbb{Z}$. In $(0,24 \pi)$, there are 20 distinct solutions.