Question 1. Let $z \in C, |z|=1$ but $z \neq -1$, then $z=\frac{1+t i}{1-t i}(t$ is a real number related to $z)$.
Given $|z|=1, z \neq-1$, hence $z=\cos \theta+i \sin \theta$, $\theta \neq k \pi$ (where $k$ is $\pm 1, \pm 3, \cdots$). Therefore, if we let $t=\operatorname{tg} \frac{\theta}{2}$, then we have
$$
\begin{aligned}
z & =\frac{1-\operatorname{tg}^{2} \frac{\theta}{2}}{1+\operatorname{tg}^{2} \frac{\theta}{2}}+i \frac{2 \operatorname{tg} \frac{\theta}{2}}{1+\operatorname{tg}^{2} \frac{\theta}{2}} \\
& =\frac{1-t^{2}+2 t i}{1+t^{2}}=\frac{(1+t i)^{2}}{(1+t i)(1-t i)} \\
& =\frac{1+t i}{1-t i} .
\end{aligned}
$$