tożsamość z modułem

[mol_ksiazkowy]

Udowodnić tożsamość:
\(\displaystyle{ \frac{|1-z^2|^2 - | z - \overline{z}|^2}{|1+z^2|^2 + | z - \overline{z}|^2}= (\frac{1-|z|^2}{1+|z|^2})^2}\)
dla \(\displaystyle{ z \in C}\)

[kicaj]

\(\displaystyle{ |1-z^2|^2 -|z-\bar{z}|^2 =(1-z^2)(1-\bar{z}^2 ) -(z-\bar{z} )(\bar{z} -z ) =1 -z^2 -\bar{z}^2 +|z|^4+z^2 -2 z\bar{z} +\bar{z}^2 =|z|^4 -2|z|^2 +1 =(|z|^2 -1)^2}\)
\(\displaystyle{ |1+z^2|^2 +|z-\bar{z}|^2 =(1+z^2)(1+\bar{z}^2 ) +(z-\bar{z} )(\bar{z} -z ) =1 +z^2 +\bar{z}^2 +|z|^4-z^2 +2 z\bar{z} -\bar{z}^2 =|z|^4 +2|z|^2 +1 =(|z|^2 +1)^2}\)