Complex Number Sum

\hspace{-16}$Here $\mathbf{\cos \alpha+\cos \beta+\cos \gamma=0..............(1)}$\\\\ \hspace{+8} and $\mathbf{\sin \alpha+\sin \beta+\sin \gamma=0................(2)\times i}$\\\\ Now Add equation $\mathbf{..(1)+(2)\;,}$ We Get\\\\ $\mathbf{(\cos \alpha+i.\sin \alpha)+(\cos \beta+i.\sin \beta)+(\cos \gamma+i.\sin \gamma)=0}$\\\\ $\mathbf{e^{i\alpha}+e^{i\beta}+e^{i\gamma}=0}$\\\\ $\mathbf{(e^{i\alpha}+e^{i\beta}+e^{i\gamma})^2=0}$\\\\ $\mathbf{e^{2i\alpha}+e^{2i\beta}+e^{2i\gamma}+2.\left\{e^{i.(\alpha+\beta)}+e^{i.(\beta+\gamma)}+e^{i.(\gamma+\alpha)}\right\}}=0$\\\\ $\mathbf{e^{2i\alpha}+e^{2i\beta}+e^{2i\gamma}+2.\left\{e^{i.(\pi-\gamma)}+e^{i.(\pi-\alpha)}+e^{i.(\pi-\beta)}\right\}}=0$\\\\ $\mathbf{e^{2i\alpha}+e^{2i\beta}+e^{2i\gamma}=0}$\\\\ $\mathbf{(\cos 2\alpha+i.\sin 2\alpha)+(\cos 2\beta+i.\sin 2\beta)+(\cos 2\gamma+i.\sin 2\gamma)=0=0+i.0}$\\\\ After equaing, We Get\\\\ $\cos 2\alpha+\cos 2\beta+\cos 2\gamma = 0$\\\\ $\sin 2\alpha+\sin 2\beta+\sin 2\gamma = 0$\\\\

How (cosα + i.sinα )= e^iα ???

take f(x)= cos x+ i sin x,
f'(x)= -sin x+ i cos x= i f(x)
now solve this diff eq