Question-131027

Question Number 131027 by Algoritm last updated on 31/Jan/21

Answered by mathmax by abdo last updated on 31/Jan/21

cos^(2m) x =(((e^(ix) +e^(−ix) )/2))^(2m) =(1/2^(2m) )Σ_(k=0) ^(2m) C_(2m) ^k (e^(ix) )^k (e^(−ix) )^((2m−k)) =(1/2^(2m) )Σ_(m=0) ^(2m) C_(2m) ^k e^(ikx) e^(−(2m−k)ix) =(1/2^(2m) )Σ_(k=0) ^(2m) C_(2m) ^k e^((ik−2mi+ik)x) =(1/2^(2m) )Σ_(k=0) ^(2m) C_(2m) ^k e^((2k−2m)ix) ⇒2^(2m) cos^(2m) x =Σ_(k=0) ^(2m ) C_(2m) ^k {cos(2k−2m)x +isin(2k−2m)x} but 2^(2m) cos^(2m) x is real ⇒ 2^(2m) cos^(2m) x =Σ_(k=0) ^(2m) C_(2m) ^k cos(2m−2k)x and Σ_(k=0) ^(2m) C_(2m) ^(k ) sin(2k−2m)x =0

$$\mathrm{cos}^{\mathrm{2m}} \mathrm{x}\:=\left(\frac{\mathrm{e}^{\mathrm{ix}} +\mathrm{e}^{−\mathrm{ix}} }{\mathrm{2}}\right)^{\mathrm{2m}} \:=\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2m}} }\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{2m}} \:\mathrm{C}_{\mathrm{2m}} ^{\mathrm{k}} \:\left(\mathrm{e}^{\mathrm{ix}} \right)^{\mathrm{k}} \left(\mathrm{e}^{−\mathrm{ix}} \right)^{\left(\mathrm{2m}−\mathrm{k}\right)} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2m}} }\sum_{\mathrm{m}=\mathrm{0}} ^{\mathrm{2m}} \:\mathrm{C}_{\mathrm{2m}} ^{\mathrm{k}} \:\mathrm{e}^{\mathrm{ikx}} \:\mathrm{e}^{−\left(\mathrm{2m}−\mathrm{k}\right)\mathrm{ix}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2m}} }\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{2m}} \:\mathrm{C}_{\mathrm{2m}} ^{\mathrm{k}} \:\mathrm{e}^{\left(\mathrm{ik}−\mathrm{2mi}+\mathrm{ik}\right)\mathrm{x}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2m}} }\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{2m}} \:\mathrm{C}_{\mathrm{2m}} ^{\mathrm{k}} \:\mathrm{e}^{\left(\mathrm{2k}−\mathrm{2m}\right)\mathrm{ix}} \:\Rightarrow\mathrm{2}^{\mathrm{2m}} \:\mathrm{cos}^{\mathrm{2m}} \mathrm{x}\:=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{2m}\:} \:\mathrm{C}_{\mathrm{2m}} ^{\mathrm{k}} \:\left\{\mathrm{cos}\left(\mathrm{2k}−\mathrm{2m}\right)\mathrm{x}\right. \\ $$$$\left.+\mathrm{isin}\left(\mathrm{2k}−\mathrm{2m}\right)\mathrm{x}\right\}\:\:\mathrm{but}\:\mathrm{2}^{\mathrm{2m}} \:\mathrm{cos}^{\mathrm{2m}} \mathrm{x}\:\mathrm{is}\:\mathrm{real}\:\Rightarrow \\ $$$$\mathrm{2}^{\mathrm{2m}} \:\mathrm{cos}^{\mathrm{2m}} \mathrm{x}\:=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{2m}} \:\mathrm{C}_{\mathrm{2m}} ^{\mathrm{k}} \:\mathrm{cos}\left(\mathrm{2m}−\mathrm{2k}\right)\mathrm{x}\:\:\mathrm{and} \\ $$$$\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{2m}} \:\mathrm{C}_{\mathrm{2m}} ^{\mathrm{k}\:} \mathrm{sin}\left(\mathrm{2k}−\mathrm{2m}\right)\mathrm{x}\:=\mathrm{0} \\ $$

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