5. (5 points) Solve the inequality $\log _{\cos x}\left(x^{2}-6 x-2\right)>\frac{2}{\log _{5} \cos x}$.
# Solution:

o.D.Z

\[
\left\{\begin{array}{c}
x^{2}-6 x-2>0 \Leftrightarrow x \in(-\infty, 3-\sqrt{11}) \cup(3+\sqrt{11},+\infty) \\
0 < \frac{2}{\log _{5} \cos x} \Leftrightarrow \log _{\cos x}\left(x^{2}-6 x-2\right)>\log _{\cos x} 25 \Leftrightarrow(c \text{ considering O.D.Z.})
\end{array}\right.
\]

![](https://cdn.mathpix.com/cropped/2024_05_06_f8af2824a37d1a011ad7g-4.jpg?height=160&width=1030&top_left_y=1136&top_left_x=867)

\[
x^{2}-6 x-2<25 \Leftrightarrow x \in(-3 ; 9).
\]

Checking O.D.Z.:

Answer: \( x \in\left(-\frac{\pi}{2}, 3-\sqrt{11}\right) \cup\left(3+\sqrt{11}, \frac{5 \pi}{2}\right) \).