Computing $(1+\cos \alpha +i\sin \alpha )^{100}$ How to prove that
$$ ‎‎‎‎‎‎‎‎‎‎‎(1+‎\cos ‎‎\alpha ‎+i‎\sin ‎‎\alpha ‎)^{100} =‎ ‎2^{100}‎\left( ‎‎\cos \left(‎\frac{‎\alpha‎}{2}\right)‎\right) ‎^{100} ‎‎\left( ‎‎\cos \left(‎\frac{100‎\alpha‎}{2}\right)+i‎\sin   \left(‎\frac{100‎\alpha‎}{2}\right)‎\right)‎‎$$
I just need a hint. I tried to write $1+‎\cos ‎‎\alpha ‎+i‎\sin ‎‎\alpha$ in polar form and use De,Moivre theorem. But it was impossible to compute $\arctan \frac{\sin \alpha}{1+\cos \alpha}$.    

$$1+\cos\alpha+i\sin \alpha=1+e^{i\alpha}=e^{\frac{i\alpha}{2}}(e^{-\frac{i\alpha}{2}}+e^{\frac{i\alpha}{2}})=2\cos\left(\frac{\alpha}{2}\right)e^{\frac{i\alpha}{2}}$$
