**Case 1:**
I noticed that the parametrization of $x=\cosh(t)$ and $y=\sinh(t)$ **only** gives us the righthand side of the graph of $x^2-y^2=1$, not the lefthandside. 

I tried to de-parametrize (or whatever that is called) by squaring: $x^2=\cosh^2(t)$ and $y^2=\sinh^2(t)$. Therefore $\cosh^2(t)-\sinh^2(t)=x^2-y^2=1$. Of course, left-hand-side is introduced because of the squaring.

**Case 2:**
Now I wondered about the parametrization of $x=\cos(t)$ and $y=\sin(t)$. By squaring we get: $\cos^2(t)+\sin^2(t)=x^2+y^2=1$

Question: why doesn't squaring in the second case introduce new parts to the graph? In other words, why does $x^2-y^2=1$ give us more values than the underlying parametrization, while $x^2+y^2=1$ does not?