$5.47 \operatorname{tg}^{4} x+\operatorname{ctg}^{4} x+\operatorname{tg}^{2} x-\operatorname{ctg}^{2} x=\frac{106}{9}$. 

Note: In this context, $\operatorname{tg}$ and $\operatorname{ctg}$ are the notations for tangent (tan) and cotangent (cot) respectively. The equation can be rewritten using more common notations as:

$5.47 \tan^{4} x+\cot^{4} x+\tan^{2} x-\cot^{2} x=\frac{106}{9}$.
5.47 Instruction. Put $\operatorname{tg}^{2} x-\operatorname{ctg}^{2} x=z$; then $\operatorname{tg}^{4} x+$ $+\operatorname{ctg}^{4} x=z^{2}+2$.

Answer: $x_{1}=\frac{\pi}{3}(3 k \pm 1), x_{2}= \pm \frac{1}{2} \arccos \frac{\sqrt{157}-6}{11}+\pi k, k \in Z$.