3. Solve the inequality

$$
(1+\sqrt{3}) \sin 2 x+2 \cos ^{2} x \geq 2\left(1+\sqrt{3} \cos ^{2} x\right)
$$
Solution. Using the formula $2 \cos ^{2} x=1+\cos 2 x$, we rewrite the given inequality as

$$
(1+\sqrt{3}) \sin 2 x \geq 1+\sqrt{3}+(\sqrt{3}-1) \cos 2 x
$$

Let $t=\cos 2 x$. If $x$ is a solution to the given inequality, then $\sin 2 x \geq 0$, so $\sin 2 x=\sqrt{1-t^{2}}$ and

$$
(1+\sqrt{3}) \sqrt{1-t^{2}} \geq 1+\sqrt{3}+(\sqrt{3}-1) t
$$

We can square inequality (2) since both of its sides are non-negative, which gives us the inequality $(2 t+1) t \leq 0$, whose solution is the set $\left[-\frac{1}{2}, 0\right]$. The set of solutions to inequality (1) is

$$
S=\left\{x \left\lvert\,-\frac{1}{2} \leq \cos 2 x \leq 0\right., \sin 2 x \geq 0\right\}=\underset{k=-\infty}{+\infty}\left[\frac{\pi}{4}+k \pi, \frac{\pi}{3}+k \pi\right]
$$