# $\sum \cos$ when angles are in arithmetic progression [duplicate]

Prove $$\cos(\alpha) + \cos(\alpha + \beta) + \cos(\alpha + 2\beta) + \dots + \cos[\alpha + (n-1)\beta] = \frac{\cos(\alpha + \frac{n-1}{2}\beta) \cdot \sin\frac{n\beta}{2}}{\sin\frac{\beta}{2}}$$

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## marked as duplicate by Aryabhata, Did, t.b., Américo Tavares, QuixoticMar 6 '12 at 20:38

$\cos(\alpha)+\cos(\alpha+\beta)+...+\cos(\alpha+(n-1)\beta)={\displaystyle \dfrac{1}{2}\sum_{k=0}^{n-1}e^{i(\alpha+k\beta)}+e^{-i(\alpha+k\beta)}=\dfrac{1}{2}}\left(e^{i\alpha}\dfrac{e^{in\beta}-1}{e^{i\beta}-1}+e^{-i\alpha}\dfrac{e^{-in\beta}-1}{e^{-i\beta}-1}\right)=\dfrac{1}{2}\left(e^{i(\alpha+\dfrac{n-1}{2}\beta)}\dfrac{e^{i\dfrac{n}{2}\beta}-e^{-i\dfrac{n}{2}\beta}}{e^{i\dfrac{1}{2}\beta}-e^{-i\dfrac{1}{2}\beta}}+e^{-i(\alpha+\dfrac{n-1}{2}\beta)}\dfrac{e^{-i\dfrac{n}{2}\beta}-e^{i\dfrac{n}{2}\beta}}{e^{-i\dfrac{1}{2}\beta}-e^{i\dfrac{1}{2}\beta}}\right)=$
$\dfrac{e^{i(\alpha+\dfrac{n-1}{2}\beta)}+e^{-i(\alpha+\dfrac{n-1}{2}\beta)}}{2}\left(\dfrac{e^{i\dfrac{n}{2}\beta}-e^{-i\dfrac{n}{2}\beta}}{e^{i\dfrac{1}{2}\beta}-e^{-i\dfrac{1}{2}\beta}}\right)=\dfrac{\cos\left(\alpha+\dfrac{n-1}{2}\beta\right)\sin\left(\dfrac{n\beta}{2}\right)}{\sin\left(\dfrac{\beta}{2}\right)}$