Question Number 73056 by mathmax by abdo last updated on 05/Nov/19
$${if}\:\left({xsina}+{cosa}\right)^{{n}} ={q}\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)+{r}\:{find}\:{r} \\ $$
Commented by mathmax by abdo last updated on 06/Nov/19
$${degr}<\mathrm{2}\:\Rightarrow{r}\left({x}\right)=\alpha{x}\:+\beta\:\Rightarrow\left({cosa}\:+{xsina}\right)^{{n}} \:={q}\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)+\alpha{x}+\beta \\ $$$${x}={i}\:\Rightarrow\left({cosa}+{isina}\right)^{{n}} \:=\alpha{i}\:+\beta\:\Rightarrow{cos}\left({na}\right)+{isin}\left({na}\right)=\alpha{i}\:+\beta \\ $$$${x}=−{i}\:\Rightarrow\left({cosa}−{isina}\right)^{{n}} =−\alpha{i}+\beta\:\Rightarrow{cos}\left({na}\right)−{isin}\left({a}\right)\:=−\alpha{i}\:+\beta\:\Rightarrow \\ $$$$\Rightarrow\mathrm{2}{cos}\left({na}\right)\:=\mathrm{2}\beta\:\Rightarrow\beta\:={cos}\left({na}\right)\:{and}\:\mathrm{2}{isin}\left({na}\right)=\mathrm{2}\alpha{i}\:\Rightarrow \\ $$$$\alpha={sin}\left({na}\right)\:\Rightarrow{r}\left({x}\right)={sin}\left({na}\right){x}\:+{cos}\left({na}\right) \\ $$