How would you prove that z - 1/z = 2*j*sin(theta) ?
I don't understand if you meant comment for me or for the OP.
As I said it is true if $\displaystyle |z|=1$ it is true. Here is the standard proof.
Recall that if $\displaystyle z=\text{c}i\text{s}(\theta)$ then $\displaystyle \frac{1}{z}=\text{c}i\text{s}(-\theta)$.
Now it is just a exercise to finish.