Question Number 191610 by mehdee42 last updated on 27/Apr/23
$${Q}:\:{if}\:\:{x}+\frac{\mathrm{1}}{{x}}=\mathrm{2}{cos}\left(\theta\right)\:\:{prove}\:{it}\:\:{x}^{{n}} +\frac{\mathrm{1}}{{x}^{{n}} }=\mathrm{2}{cos}\left({n}\theta\right) \\ $$
Answered by Frix last updated on 27/Apr/23
$${z}+\frac{\mathrm{1}}{{z}}=\mathrm{2}{c}\:\Rightarrow\:{z}={c}\pm\sqrt{{c}^{\mathrm{2}} −\mathrm{1}}\:\Rightarrow \\ $$$${x}=\mathrm{cos}\:\theta\:\pm\mathrm{i}\:\mathrm{sin}\:\theta\:=\mathrm{e}^{\pm\mathrm{i}\theta} \\ $$$${x}^{{n}} +\frac{\mathrm{1}}{{x}^{{n}} }=\mathrm{e}^{\pm\mathrm{i}{n}\theta} +\mathrm{e}^{\mp\mathrm{i}{n}\theta} =\mathrm{2cos}\:{n}\theta \\ $$
Commented by mehdee42 last updated on 27/Apr/23
$${very}\:{nice}\:{sir} \\ $$