Task B-4.6. Solve the inequality

$$
0.2^{\cos 2 x}-\frac{1}{25^{\cos ^{2} x}}>4 \cdot 125^{-\frac{1}{2}}, \quad x \in[0,2 \pi]
$$
## Solution.

$$
\begin{aligned}
5^{-\cos 2 x}-5^{-2 \cos ^{2} x} & >4 \cdot 5^{-\frac{3}{2}} \\
5^{1-2 \cos ^{2} 2 x}-5^{-2 \cos ^{2} x} & >4 \cdot 5^{-\frac{3}{2}} \\
5^{-2 \cos ^{2} x} \cdot(5-1) & >4 \cdot 5^{-\frac{3}{2}} \\
5^{-2 \cos ^{2} x} & >5^{-\frac{3}{2}} \\
-2 \cos ^{2} x & >-\frac{3}{2} \\
\cos ^{2} x & <\frac{3}{2} \\
|\cos x| & <\frac{\sqrt{3}}{2}
\end{aligned}
$$

The final solution is $x \in\left\langle\frac{\pi}{6}, \frac{5 \pi}{6}\right\rangle \cup\left\langle\frac{7 \pi}{6}, \frac{11 \pi}{6}\right\rangle$.