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Question Number 28259 by abdo imad last updated on 22/Jan/18
$${let}\:{give}\:\:\:\:{A}=\:\:\:\left(\:\:{cos}\theta\:\:\:\:\:\:\:−{sin}\theta\:\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\:{sin}\theta\:\:\:\:\:\:\:\:\:{cos}\theta\:\:\:\:\right) \\ $$$$\left.\mathrm{1}\right)\:\:{calculate}\:^{{t}} {A}.\:{A}\:\:.{prove}\:{that}\:{A}\:{is}\:{inversible}\:{and}\:{find} \\ $$$${A}^{−\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:\:{find}\:\:{A}^{{n}} \:\:\:{for}\:{n}\in\:{N} \\ $$$$\left.\mathrm{3}\right)\:{developp}\:\left({A}\:+{A}^{−\mathrm{1}} \right)^{{n}} \:\:{then}\:{prove}\:{that} \\ $$$$\mathrm{2}^{{n}} \:{cos}^{{n}} \theta\:\:=\:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \left({n}−\mathrm{2}{k}\right)\theta\:\:{and} \\ $$$$\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{n}} \:{sin}\left({n}−\mathrm{2}{k}\right)\theta\:\:=\mathrm{0}\:\:. \\ $$