Parametrize the saddle surface

Question

How do I show that Ψ is one-to-one?

Answer 1

Hints:- $$ Ψ(u_1,v_1)= Ψ(u_2,v_2)$$ $\implies u_1\cosh v_1=u_2\cosh v_2$ and $u_1\cosh v_1=u_2\cosh v_2$ and $u_1^2=u_2^2$ $$\implies \frac{u_1}{u_2}=\frac{\sinh v_1}{\sinh v_2}=\frac{\cosh v_1}{\cosh v_2} $$

$$\implies \sinh (v_1-v_2)=0$$