44. Solve the inequality $4 \log _{16} \cos 2 x+2 \log _{4} \sin x+\log _{2} \cos x+$ $+3<0$.
44. We transition to logarithms with base 2. The given inequality reduces to the inequality $\log _{2} 2 \sin 4 x<0$, i.e., $0<\sin 4 x<\frac{1}{2}$.

But $0<x<\frac{\pi}{4}$. Therefore, the inequality holds under the condition $0<4 x<\frac{\pi}{6}$, i.e., $0<x<\frac{\pi}{24}$ and $\frac{5}{6} \pi<4 x<\pi$, i.e., $\frac{5}{24} \pi<x<\frac{\pi}{4}$.