Existence of a fundamental unit satisfying a Trace identity for cyclic cubic number fields?

Question

Let $K$ be a cyclic cubic number field.

1. If $w$ is a fundamental unit of negative norm, then must Trace$(w+1/w)=3$?

and a weaker question...

2. Is Trace$(w+1/w)=3$ necessarily true for some $w=uv^2$ where $u$ is a fundamental unit of negative norm and $v$ any unit?

Remarks: That $K/\mathbb{Q}$ is Galois is implied by the terminology "cyclic" and such a number field is totally real. By results of Armitage and Frohlich, every totally positive unit is a square if the class number of K is odd. If this helps, go ahead and assume odd class number.