$8.459 \cos ^{2} x^{2}\left(\operatorname{tg} x^{2}+2 \operatorname{tg} x\right)+\operatorname{tg}^{3} x \cdot\left(1-\sin ^{2} x^{2}\right)\left(2-\operatorname{tg} x \cdot \operatorname{tg} x^{2}\right)=0$.

$8.459 \cos ^{2} x^{2}\left(\tan x^{2}+2 \tan x\right)+\tan^{3} x \cdot\left(1-\sin ^{2} x^{2}\right)\left(2-\tan x \cdot \tan x^{2}\right)=0$.
Solution. The original equation is equivalent to

$\operatorname{tg} x^{2}+2 \operatorname{tg} x+2 \operatorname{tg}^{3} x-\operatorname{tg}^{4} x \cdot \operatorname{tg} x^{2}=0 \Leftrightarrow \operatorname{tg}^{2}\left(1-\operatorname{tg}^{4} x\right)+2 \operatorname{tg} x \cdot\left(1+\operatorname{tg}^{2} x\right)=0 \Leftrightarrow$ $\Leftrightarrow \operatorname{tg} x^{2}\left(1-\operatorname{tg}^{2} x\right)=-2 \operatorname{tg} x \Leftrightarrow \operatorname{tg} x^{2}=-\frac{2 \operatorname{tg} x}{1-\operatorname{tg}^{2} x} \Leftrightarrow\left\{\begin{array}{l}\operatorname{tg} x^{2}+\operatorname{tg} 2 x=0 \\ \cos x \neq 0\end{array} \Leftrightarrow\right.$

$\Leftrightarrow\left\{\begin{array}{l}\frac{\sin \left(x^{2}+2 x\right)}{\cos x^{2} \cdot \cos 2 x} \\ \cos x \neq 0\end{array}=0 \Leftrightarrow\left\{\begin{array}{l}x^{2}+2 x=\pi k, \\ x^{2} \neq \frac{\pi}{2}+\pi n, \\ x \neq \frac{\pi}{4}+\frac{\pi m}{2}, \\ x \neq \frac{\pi}{2}+\pi l\end{array} \Leftrightarrow\left\{\begin{array}{l}(x+1)^{2}=\pi k+1, \\ x^{2} \neq \frac{\pi}{2}+\pi n, \\ x \neq \frac{\pi}{4}+\frac{\pi m}{2}, \\ x \neq \frac{\pi}{2}+\pi l\end{array} \Leftrightarrow\right.\right.\right.$

$\Leftrightarrow\left\{\begin{array}{l}x=-1 \pm \sqrt{\pi k+1}, k=0,1,2, \ldots, \\ x^{2} \neq \frac{\pi}{2}+\pi n, \\ x \neq \frac{\pi}{4}+\frac{\pi m}{2}, \\ x \neq \frac{\pi}{2}+\pi l .\end{array}\right.$

The last three inequalities in the system are satisfied when $x=-1 \pm \sqrt{\pi k+1}$.

Answer: $\quad x=-1 \pm \sqrt{\pi k+1}, k=0,1,2, \ldots$.