Calculations with complex numbers So I encountered this one question today in my math book, and I don't know how to get the right answer even though it seems really easy, I just wanna know how to do it so i can get some sleep.
This is the question:

When the complex number $z=r(\cos\theta + i \sin\theta)$ show that $z-\frac{1}{z}=i(2r\sin\theta)$

I haven't done much for this question other than do this using the actual complex number:
$$\frac{z^2-1}{z}$$
Can anyone show me how to do prove this question, any help is appreciated.

What's equal to $\:2ir\sin\theta\:$ is $\:z -\color{red}{\bar z}$.
Now $\bar z=\dfrac 1 z\:$ if and only if $z$ has modulus $1$.
