2. Solve the equation $8 \sin (x)+12 \sin ^{3}(x)+2022 \sin ^{5}(x)=8 \cos (2 x)+$ $12 \cos ^{3}(2 x)+2022 \cos ^{5}(2 x)$
Solution. Let's introduce the function $f(t)=8 t+12 t^{3}+2022 t^{5}$, and then the original equation can be represented as $f(\sin (x))=f(\cos (2 x))$.

Note that the introduced function $f(t)$ is monotonically increasing (as the sum of three monotonically increasing functions). The monotonicity of the function $f(t)$ can also be easily verified by examining the derivative of this function.

Since $f(t)$ is monotonic, the equation $f(\sin (x))=f(\cos (2 x))$ is equivalent to the equation

$$
\sin (x)=\cos (2 x)
$$

From the last equation, we get $2 \sin ^{2}(x)+\sin (x)-1=0$, from which $\sin (x)=-1$ or $\sin (x)=\frac{1}{2}$.

Thus, $x=-\frac{\pi}{2}+2 \pi k, x=(-1)^{k} \frac{\pi}{6}+\pi k, k \in Z$.

## Recommendations for evaluating solutions.

If it is proven that the original equation is equivalent to the equation $\sin (x)=$ $\cos (2 x)$ - no less than 4 points.

If only the solutions of the equation $\sin (x)=\cos (2 x)$ are correctly found, but it is not proven that there are no other solutions - 2 points.