Consider the imaginary

Prove that the following identity holds:

If the quantity is positive, prove that the imaginary parts of the complex numbers and have the same sign.

*Proof.* For the first part of the proof we compute using the definition of and the fact that .

This proves the first part of the theorem. To prove the second part, first we note that for any complex numbers and we have

Therefore, from the identity we established in the first part we have,

So, if then the quantity

(since the denominator is always positive as well). Therefore, and have the same sign