  Given a prime power $q$ and a positive integer $n$, let $\mathbb{F}_{q^{n}}$
denote the finite field with $q^n$ elements. Also let $a,b$ be arbitrary
members of the ground field $\mathbb{F}_{q}$. We investigate the existence of a
non-zero element $\xi \in \mathbb{F}_{q^{n}}$ such that $\xi+ \xi^{-1}$ is
primitive and $T(\xi)=a, T(\xi^{-1})=b$, where $T(\xi)$ denotes the trace of
$\xi$ in $\mathbb{F}_{q}$. This was a question intended to be addressed by Cao
and Wang in 2014. Their work dealt instead with another problem already in the
literature. Our solution deals with all values of $n \geq 5$.
  A related study involves the cubic extension $\mathbb{F}_{q^{3}}$ of
$\mathbb{F}_{q}$. We show that if $q\geq 8\cdot 10^{12}$ then, for any $a\in
\mathbb{F}_{q}$ we can find a primitive element $\xi \in \mathbb{F}_{q^{3}}$
such that $\xi + \xi^{-1}$ is also a primitive element of $\mathbb{F}_{q^{3}}$,
and for which the trace of $\xi$ is equal to $a$. The improves a result of
Cohen and Gupta. Along the way we prove a hybridised lower bound on prime
divisors in various residue classes, which may be of interest to related
existence questions.
