1. Can one of the angles of a triangle ( $x$ ) satisfy the equation: $\cos \frac{x}{2}=\cos \frac{x}{3} ?$
Solution. Let's find the possible values of $x$ from the given equation. For this, we will transform it:

$\cos \frac{x}{2}-\cos \frac{x}{3}=-2 \sin \frac{5 x}{12} \sin \frac{x}{12}=0$

Then the possible cases are:

$\sin \frac{5 x}{12}=0$ or $\frac{5 x}{12}=n \pi, n \in \boldsymbol{Z}$ and the smallest positive value of $x=\frac{12}{5} \pi>\pi$, which is impossible for an angle of a triangle.

$\sin \frac{x}{12}=0$ or $\frac{x}{12}=n \pi, n \in \boldsymbol{Z}$ and the smallest positive value of $x=12 \pi>\pi$, which is also impossible for an angle of a triangle.

Answer: cannot.