2. Solve the equation $2 \log _{3} \operatorname{ctg} x=\log _{2} \cos x$.
Solution 1. Rewrite the equation as

$$
\log _{3}(\operatorname{ctg} x)=\log _{4}(\cos x)=t
$$

Then $\cos x=4^{t}, \operatorname{ctg} x=3^{t}$ and $\sin x=\left(\frac{4}{3}\right)^{t}$, so $4^{2 t}+\left(\frac{4}{3}\right)^{2 t}=1$. The function on the right side of the obtained equation is increasing, so it has no more than one solution. It is clear that the solution is $t=-\frac{1}{2}$, from which $x=\frac{\pi}{3}+2 \pi k, k \in \mathbb{Z},($ since $\sin x>0)$.

Answer: $x=\frac{\pi}{3}+2 \pi k, k \in \mathbb{Z}$.