Does anyone know how to prove this?
Let $\displaystyle z \neq -1$ be complex number, which modulo is 1. Prove that it can be expressed in the form: $\displaystyle z = \frac{1+ti}{1-ti}$ where t is real number
Why would you let z be that value? Your told |z| = 1...
Going on from what you have put, if $\displaystyle \displaystyle z = \frac{1 - t^2}{1 + t^2} + \frac{2t}{1 + t^2}\,i$, all that is left is to show its modulus is 1.
$\displaystyle \displaystyle \begin{align*} |z| &= \sqrt{ \left( \frac{1 - t^2}{1 + t^2} \right) ^2 + \left( \frac{2t}{1 + t^2} \right) ^2 } \\ &= \sqrt{ \frac{1 - 2t^2 + t^4 + 4t^2}{\left( 1 +t^2 \right) ^2} } \\ &= \sqrt{ \frac{1 + 2t^2 + t^4}{1 + 2t^2 + t^4} } \\ &= \sqrt{1} \\ &= 1 \end{align*}$