8.309. $\operatorname{tg}\left(t^{2}-t\right) \operatorname{ctg} 2=1$.

8.309. $\tan\left(t^{2}-t\right) \cot 2=1$.
## Solution.

Domain of definition: $\cos \left(t^{2}-t\right) \neq 0$.

From the condition we have

$\operatorname{tg}\left(t^{2}-t\right)=\operatorname{tg} 2, \operatorname{tg}\left(t^{2}-t\right)-\operatorname{tg} 2=0 \Leftrightarrow \frac{\sin \left(t^{2}-t-2\right)}{\cos \left(t^{2}-t\right) \cos 2}=0 \Leftrightarrow$

$\Leftrightarrow \sin \left(t^{2}-t-2\right)=0 \Leftrightarrow t^{2}-t-2=\pi k, k \in Z, t^{2}-t-2-\pi k=0, k \in Z$.

From this equation we obtain

$t_{1,2}=\frac{1 \pm \sqrt{9+4 \pi k}}{2}$, where $9+4 \pi k \geq 0, k \geq-\frac{9}{4 \pi}, k=0 ; 1 ; 2 ; \ldots$.

Answer: $t_{1,2}=\frac{1 \pm \sqrt{9+4 \pi k}}{2}$, where $k=0 ; 1 ; 2 ; \ldots$.