Question Number 127558 by mohammad17 last updated on 30/Dec/20
$${if}\:{Z}={cos}\theta+{isin}\theta\:{prove}\:{that}\: \\ $$$$ \\ $$$${Z}^{\mathrm{5}} −\frac{\mathrm{1}}{{Z}^{\mathrm{2}} }=\mathrm{2}{isin}\mathrm{5}\theta\: \\ $$$$ \\ $$$${help}\:{me}\:{sir} \\ $$
Commented by mohammad17 last updated on 30/Dec/20
$$? \\ $$
Commented by MJS_new last updated on 31/Dec/20
$${z}^{\mathrm{5}} +\frac{\mathrm{1}}{{z}^{\mathrm{2}} }=\mathrm{cos}\:\mathrm{5}\theta\:−\mathrm{cos}\:\mathrm{2}\theta\:+\mathrm{i}\:\left(\mathrm{sin}\:\mathrm{5}\theta\:+\mathrm{sin}\:\mathrm{2}\theta\right) \\ $$
Answered by mohammad17 last updated on 30/Dec/20
$${how}\:{can}\:{solve}\:{this} \\ $$
Answered by Dwaipayan Shikari last updated on 30/Dec/20
$${z}^{\mathrm{5}} −{z}^{−\mathrm{5}} ={e}^{\mathrm{5}{i}\theta} −{e}^{−\mathrm{5}{i}\theta} =\mathrm{2}{isin}\mathrm{5}\theta \\ $$
Commented by mohammad17 last updated on 30/Dec/20
$${put}\:{sir}\:{how}\:{became}\:{z}^{−\mathrm{5}} {can}\:{you}\:{give}\:{me}\:{stebs} \\ $$
Commented by Dwaipayan Shikari last updated on 30/Dec/20
$${I}\:{think}\:{mistake}\:{in}\:{question}\:.{It}\:{should}\:{be}\:{z}^{−\mathrm{5}} \:{instead}\:{of}\:{z}^{−\mathrm{2}} \\ $$