Question

Prove that $cos\alpha +cos(\alpha +\beta )+cos(\alpha +2\beta )+.....+cos(\alpha +(n-1\left)\beta \right)$

$=\frac{cos\{\alpha +\frac{n-1}{2}\beta \}sin\left(\frac{n\beta }{2}\right)}{sin\frac{\beta }{2}}$ for all n $ϵ$ N.

Answer 1

Let C = $cos\alpha +cos(\alpha +\beta )+.....+cos(\alpha +(n-1\left)\beta \right)$

$s=sin\alpha +sin(\alpha +\beta )+......+sin(\alpha +(n-1\left)\beta \right)$

Consider $C+iS=e^{i\alpha }+e^{i(\alpha +\beta )+.....+e^{i(\alpha +(n-1\left)\beta \right)}}wherei=\sqrt{-1}$

C + iS =$e^{i\alpha }[1+e^{i\beta }+e^{i2\beta }+...+e^{i(n-1)\beta }]$

$=e^{i\alpha \left(\frac{e^{in\beta -1}}{e^{i\beta }-1}\right)}$

$C+iS=e^{i(\alpha +(n-1\left)\frac{\beta }{2}\right)}\frac{sin\left(\frac{n\beta }{2}\right)}{sin\left(\frac{\beta }{2}\right)}$

Real term is

$C=cos\alpha +cos(\alpha +\beta )+....+cos(\alpha +(n-1\left)\beta \right)=\frac{cos\{\alpha +(n-1\left)\frac{\beta }{2}\right\}sin\left(\frac{n\beta }{2}\right)}{sin\left(\frac{\beta }{2}\right)}$