$8.490 \tan x + \cot x + \tan^2 x + \cot^2 x + \tan^3 x + \cot^3 x = 6$.
Solution. Let $\operatorname{tg} x+\operatorname{ctg} x=y$. Then

$\operatorname{tg}^{2} x+\operatorname{ctg}^{2} x=y^{2}-2, \operatorname{tg}^{3} x+\operatorname{ctg}^{3} x=(\operatorname{tg} x+\operatorname{ctg} x)\left(\operatorname{tg}^{2} x+\operatorname{ctg}^{2} x-1\right)=y\left(y^{2}-3\right)$.

The original equation can be written as:

$y+y^{2}-2+y\left(y^{2}-3\right)=6 \Leftrightarrow y^{3}+y^{2}-2 y-8=0 \Leftrightarrow$

$\Leftrightarrow(y-2)\left(y^{2}+3 y+4\right)=0 \Leftrightarrow \operatorname{tg} x+\operatorname{ctg} x=2 \Leftrightarrow(\operatorname{tg} x-1)^{2}=0 \Leftrightarrow$

$\Leftrightarrow \operatorname{tg} x=1 \Leftrightarrow x=\frac{\pi}{4}+\pi k$.

Answer: $x=\frac{\pi}{4}(4 k+1)$.