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Question Number 76718 by peter frank last updated on 29/Dec/19
For n ∈ N prove by mathematical  induction that  cos α+cos (α+β)+cos [α+(n−1)β]+...cos [α+(n−1)β]=  ((cos [α+(((n−1)/2))β]sin ((nβ)/2))/(sin (n/2)))
$${For}\:{n}\:\in\:{N}\:{prove}\:{by}\:{mathematical} \\ $$$${induction}\:{that} \\ $$$$\mathrm{cos}\:\alpha+\mathrm{cos}\:\left(\alpha+\beta\right)+\mathrm{cos}\:\left[\alpha+\left({n}−\mathrm{1}\right)\beta\right]+…\mathrm{cos}\:\left[\alpha+\left({n}−\mathrm{1}\right)\beta\right]= \\ $$$$\frac{\mathrm{cos}\:\left[\alpha+\left(\frac{{n}−\mathrm{1}}{\mathrm{2}}\right)\beta\right]\mathrm{sin}\:\frac{{n}\beta}{\mathrm{2}}}{\mathrm{sin}\:\frac{{n}}{\mathrm{2}}} \\ $$

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