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

Hint:$$\cos z=\frac{e^{iz}+e^{-iz}}{2}$$
and $$\overline{e^{iz}}=\overline{e^{xi-y}}=\overline{\frac{e^{xi}}{e^y}}= \frac{\overline {\cos x+i\sin x}}{e^y}=\frac{e^{-xi}}{e^y}=e^{-xi-y}=e^{i(-x+iy)}=e^{-i\bar{z}}$$
Therefore, 
$$\overline{\cos z}=\frac{e^{-i\bar{z}}+e^{i\bar{z}}}{2}=\cos \bar{z}$$
