Let $\alpha \beta \in \mathbb{R}$. Prove that \[e^{i\alpha} + e^{i\beta} = 2 \cos \left(

Question

Let $\alpha > \beta \in \mathbb{R}$. Prove that $$e^{i\alpha} + e^{i\beta} = 2 \cos \left( \frac{\alpha - \beta}{2} \right) e^{i \left( \frac{\alpha + \beta}{2} \right)}.$$ [Hint: There are $3$ ways to prove this.]

Accepted Answer

$e^{i\alpha}+e^{i\beta}=2\cos\!\left(\dfrac{\alpha-\beta}{2}\right)e^{i\left(\frac{\alpha+\beta}{2}\right)}$ (proven for all real $\alpha>\beta$).

Vector sum of complex numbers via exponential representation

Vector sum of complex numbers via exponential representation

Solution:

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