$5.48 \frac{3(\cos 2 x+\operatorname{ctg} 2 x)}{\operatorname{ctg} 2 x-\cos 2 x}-2(\sin 2 x+1)=0$.

$5.49(\cos x-\sin x)^{2}+\cos ^{4} x-\sin ^{4} x=0.5 \sin 4 x$.
5.48 We have

$$
\frac{3 \cos 2 x\left(1+\frac{1}{\sin 2 x}\right)}{\cos 2 x\left(\frac{1}{\sin 2 x}-1\right)}-2(\sin 2 x+1)=0, \cos 2 x \neq 0, \sin 2 x \neq 0
$$

$$
\frac{3(\sin 2 x+1)}{1-\sin 2 x}-2(\sin 2 x+1)=0
$$

$\sin 2 x \neq -1$, since in this case $\cos 2 x=0$;

$$
\frac{3}{1-\sin 2 x}=2, \sin 2 x=-\frac{1}{2} ; 2 x=(-1)^{k+1} \frac{\pi}{6}+\pi k
$$

Solution: $x=(-1)^{k+1} \frac{\pi}{12}+\frac{\pi k}{2}, k \in Z$.