How to calculate $\overline{\cos \phi}$ How do you calculate $\overline{\cos \phi}$? Where $\phi\in\mathbb{C}$.
I try to proof that $\cos \phi \cdot \overline{\cos \phi} +\sin \phi \cdot \overline{\sin \phi}=1$?

$$\cos(a+ib)=\frac{e^{i(a+ib)}+e^{-i(a+ib)}}2$$
$$=\frac{e^{-b}(\cos a+i\sin a)+e^b(\cos  a-i\sin a)}2$$
$$=\cos a\frac{e^b+e^{-b}}2-i\sin a\frac{e^b-e^{-b}}2=\cos a\cosh b-i\sin a\sinh b$$
We know  $f(X+iY)=A+iB\iff f(X-iY)=A-iB$
So, $$\cos(a-ib)=\cos a\cosh b+i\sin a\sinh b$$
