8.383. $\log _{\cos x} 4 \cdot \log _{\cos ^{2} x} 2=1$.
## Solution.

Domain of definition: $0<\cos x<1$.

Switch to the base $\cos x$. We get

$\frac{1}{2} \log _{\cos x} 2^{2} \cdot \log _{\cos x} 2=1, \log _{\cos x}^{2} 2=1$.

From here

1) $\log _{\cos x} 2=1 \Leftrightarrow \cos x=2, \varnothing$
2) $\log _{\cos x} 2=-1, \frac{1}{\cos x}=2, \cos x=\frac{1}{2}, x= \pm \frac{\pi}{3}+2 \pi k=\frac{\pi}{3}(6 k \pm 1), k \in Z$.

Answer: $x=\frac{\pi}{3}(6 k \pm 1), k \in Z$.