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If-z-1-z-2cos-show-that-z-m-1-z-m-2cosm-




Question Number 138612 by SOMEDAVONG last updated on 15/Apr/21
If z+ (1/z) =2cosβ .show that z^m +(1/z^m ) =2cosmβ.
$$\mathrm{If}\:\mathrm{z}+\:\frac{\mathrm{1}}{\mathrm{z}}\:=\mathrm{2cos}\beta\:.\mathrm{show}\:\mathrm{that}\:\mathrm{z}^{\mathrm{m}} +\frac{\mathrm{1}}{\mathrm{z}^{\mathrm{m}} }\:=\mathrm{2cosm}\beta. \\ $$
Answered by Ar Brandon last updated on 15/Apr/21
z=e^(iβ)   z^m +z^(−m) =e^(imβ) +e^(−imβ)   =cosmβ+isinmβ+cosmβ−isinmβ  =2cos(mβ)
$$\mathrm{z}=\mathrm{e}^{\mathrm{i}\beta} \\ $$$$\mathrm{z}^{\mathrm{m}} +\mathrm{z}^{−\mathrm{m}} =\mathrm{e}^{\mathrm{im}\beta} +\mathrm{e}^{−\mathrm{im}\beta} \\ $$$$=\mathrm{cosm}\beta+\mathrm{isinm}\beta+\mathrm{cosm}\beta−\mathrm{isinm}\beta \\ $$$$=\mathrm{2cos}\left(\mathrm{m}\beta\right) \\ $$

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