8.090. $\operatorname{tg} \frac{t}{2} \operatorname{ctg} \frac{3 t}{2}+\cos ^{-1} \frac{t}{2} \sin ^{-1} \frac{3 t}{2}=1$.

8.090. $\tan \frac{t}{2} \cot \frac{3 t}{2}+\cos ^{-1} \frac{t}{2} \sin ^{-1} \frac{3 t}{2}=1$.
## Solution.

Domain of definition: $\left\{\begin{array}{l}\cos \frac{t}{2} \neq 0, \\ \sin \frac{3 t}{2} \neq 0 .\end{array}\right.$

Rewrite the equation as

$$
\begin{aligned}
& \frac{\sin \frac{t}{2} \cos \frac{3 t}{2}}{\cos \frac{t}{2} \sin \frac{3 t}{2}}+\frac{1}{\cos \frac{t}{2} \sin \frac{3 t}{2}}-1=0 \Rightarrow \sin \frac{t}{2} \cos \frac{3 t}{2}-\cos \frac{t}{2} \sin \frac{3 t}{2}+1=0 \Leftrightarrow \\
& \Leftrightarrow \sin \left(\frac{t}{2}-\frac{3 t}{2}\right)+1=0,-\sin t+1=0, \quad \sin t=1
\end{aligned}
$$

From this, $t=\frac{\pi}{2}+2 \pi k=\frac{\pi}{2}(4 k+1), k \in Z$.

Answer: $\quad t=\frac{\pi}{2}(4 k+1), \quad k \in Z$.