Prove that if $z+\frac1z$ is real, then either $|z|=1$ or $z$ is real. 
Prove that if $z+\frac1z$ is real, then either $|z|=1$ or $z$ is real. 

Original image
I am not sure whether my proof is sufficient. So far, I have shown that 
$$z+\frac1z = \frac{z^2+1}{z}=\frac{|z|+1}{z}$$
However, I don't think the proof enough, and also I seemed to have proven that they both have to be real and not either... Please advise.
Note: I realised that my proof is wrong, as it was kindly mentioned that $z^2$ doesn't equal $|z|$. I remembered wrongly, it should be $zz*=|z|$ with $z*$ being a conjugate of $z$. 

Another way:
\begin{align}
z+\frac{1}{z}&=\rho(\cos\theta+i\sin\theta)+\frac{1}{\rho}(\cos\theta-i\sin\theta)\\
&\left(\rho+\frac{1}{\rho}\right)\cos\theta+i\left(\rho-\frac{1}{\rho}\right)\sin\theta
\end{align}
And
$$
\left(\rho-\frac{1}{\rho}\right)\sin\theta=0\implies \rho=1 \text{ or } \theta=0 \text{ or } \theta=\pi
$$
