TITLE: Computing $(1+\cos \alpha +i\sin \alpha )^{100}$
QUESTION [6 upvotes]: 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 $\arctan \frac{\sin \alpha}{1+\cos \alpha}$.

REPLY [3 votes]: The figure below should make it easy to simplify the expression
$\arctan \dfrac{\sin\alpha}{1+\cos\alpha}$.
Note the isoceles triangle; you are looking for the angle at vertex $A$.