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Question Number 66462 by mathmax by abdo last updated on 15/Aug/19

1)simplify S_n (x)=Σ_(k=0) ^n C_n ^k cos^k (x)cos(kx) 2)find the value of A_n =Σ_(k=0) ^n C_n ^k cos^k ((π/n))cos(((kπ)/n))

$$\left.\mathrm{1}\right){simplify}\:{S}_{{n}} \left({x}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{cos}^{{k}} \left({x}\right){cos}\left({kx}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:{A}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{cos}^{{k}} \left(\frac{\pi}{{n}}\right){cos}\left(\frac{{k}\pi}{{n}}\right) \\ $$

Commented by mathmax by abdo last updated on 21/Aug/19

1) we have S_n (x)=Re( Σ_(k=0) ^n C_n ^k cos^k x e^(ikx) ) but Σ_(k=0) ^n C_n ^k cos^k xe^(ikx) =Σ_(k=0) ^n C_n ^k (e^(ix) cosx)^k ×1^(n−k) =(1+e^(ix) cosx)^n =(1+cosx{cosx +isinx})^n =(1+cos^2 x +isinx cosx)^n let z =1+cos^2 x +isinx cosx ⇒∣z∣=(√((1+cos^2 x)^2 +sin^2 x cos^2 x)) =(√(1+cos^4 x +2cos^2 x +(1−cos^2 x)cos^2 x)) =(√(1+cos^4 x +2cos^2 x+cos^2 x−cos^4 x))=⇒(√(1+3cos^2 x)) ⇒ z =(√(1+3cos^2 x))e^(i arctan(((sinx cosx)/(1+cos^2 x)))) we have ((sinxcosx)/(1+cos^2 x)) =((sin(2x))/(2(1+((1+cos(2x))/2)))) =((sin(2x))/(2+1+cos(2x))) =((sin(2x))/(3+cos(2x))) and 1+cos^2 x =1+((1+cos(2x))/2) =((3+cos(2x))/2) ⇒ z =(((3+cos(2x))/2))^(1/2) e^(i arctan(((sin(2x))/(3+cos(2x))))) ⇒ z^n =(((3+cos(2x))/2))^(n/2) e^(in arctan(((sin(2x))/(3+cos(2x))))) ⇒ S_n (x) =Re(z^n ) =(((3+cos(2x))/2))^(n/2) cos(n arctan(((sin(2x))/(3+cos(2x))))) 2) we have A_n =Σ_(k=0) ^n C_n ^k cos^k ((π/n)) cos(((kπ)/n)) =S_n ((π/n)) =(((3+cos(((2π)/n)))/2))^(n/2) cos(narctan(((sin(((2π)/n)))/(3+cos(((2π)/n)))))).

$$\left.\mathrm{1}\right)\:{we}\:{have}\:{S}_{{n}} \left({x}\right)={Re}\left(\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{cos}^{{k}} {x}\:{e}^{{ikx}} \right)\:{but} \\ $$$$\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{cos}^{{k}} {xe}^{{ikx}} \:=\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \left({e}^{{ix}} {cosx}\right)^{{k}} \:×\mathrm{1}^{{n}−{k}} =\left(\mathrm{1}+{e}^{{ix}} {cosx}\right)^{{n}} \\ $$$$=\left(\mathrm{1}+{cosx}\left\{{cosx}\:+{isinx}\right\}\right)^{{n}} \:=\left(\mathrm{1}+{cos}^{\mathrm{2}} {x}\:+{isinx}\:{cosx}\right)^{{n}} \\ $$$${let}\:{z}\:=\mathrm{1}+{cos}^{\mathrm{2}} {x}\:+{isinx}\:{cosx}\:\Rightarrow\mid{z}\mid=\sqrt{\left(\mathrm{1}+{cos}^{\mathrm{2}} {x}\right)^{\mathrm{2}} +{sin}^{\mathrm{2}} {x}\:{cos}^{\mathrm{2}} {x}} \\ $$$$=\sqrt{\mathrm{1}+{cos}^{\mathrm{4}} {x}\:+\mathrm{2}{cos}^{\mathrm{2}} {x}\:+\left(\mathrm{1}−{cos}^{\mathrm{2}} {x}\right){cos}^{\mathrm{2}} {x}} \\ $$$$=\sqrt{\mathrm{1}+{cos}^{\mathrm{4}} {x}\:+\mathrm{2}{cos}^{\mathrm{2}} {x}+{cos}^{\mathrm{2}} {x}−{cos}^{\mathrm{4}} {x}}=\Rightarrow\sqrt{\mathrm{1}+\mathrm{3}{cos}^{\mathrm{2}} {x}}\:\Rightarrow \\ $$$${z}\:=\sqrt{\mathrm{1}+\mathrm{3}{cos}^{\mathrm{2}} {x}}{e}^{{i}\:{arctan}\left(\frac{{sinx}\:{cosx}}{\mathrm{1}+{cos}^{\mathrm{2}} {x}}\right)} \\ $$$${we}\:{have}\:\:\frac{{sinxcosx}}{\mathrm{1}+{cos}^{\mathrm{2}} {x}}\:=\frac{{sin}\left(\mathrm{2}{x}\right)}{\mathrm{2}\left(\mathrm{1}+\frac{\mathrm{1}+{cos}\left(\mathrm{2}{x}\right)}{\mathrm{2}}\right)}\:=\frac{{sin}\left(\mathrm{2}{x}\right)}{\mathrm{2}+\mathrm{1}+{cos}\left(\mathrm{2}{x}\right)}\:=\frac{{sin}\left(\mathrm{2}{x}\right)}{\mathrm{3}+{cos}\left(\mathrm{2}{x}\right)} \\ $$$${and}\:\mathrm{1}+{cos}^{\mathrm{2}} {x}\:=\mathrm{1}+\frac{\mathrm{1}+{cos}\left(\mathrm{2}{x}\right)}{\mathrm{2}}\:=\frac{\mathrm{3}+{cos}\left(\mathrm{2}{x}\right)}{\mathrm{2}}\:\Rightarrow \\ $$$${z}\:=\left(\frac{\mathrm{3}+{cos}\left(\mathrm{2}{x}\right)}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:{e}^{{i}\:{arctan}\left(\frac{{sin}\left(\mathrm{2}{x}\right)}{\mathrm{3}+{cos}\left(\mathrm{2}{x}\right)}\right)} \:\Rightarrow \\ $$$${z}^{{n}} \:=\left(\frac{\mathrm{3}+{cos}\left(\mathrm{2}{x}\right)}{\mathrm{2}}\right)^{\frac{{n}}{\mathrm{2}}} \:{e}^{{in}\:{arctan}\left(\frac{{sin}\left(\mathrm{2}{x}\right)}{\mathrm{3}+{cos}\left(\mathrm{2}{x}\right)}\right)} \:\Rightarrow \\ $$$${S}_{{n}} \left({x}\right)\:={Re}\left({z}^{{n}} \right)\:=\left(\frac{\mathrm{3}+{cos}\left(\mathrm{2}{x}\right)}{\mathrm{2}}\right)^{\frac{{n}}{\mathrm{2}}} \:{cos}\left({n}\:{arctan}\left(\frac{{sin}\left(\mathrm{2}{x}\right)}{\mathrm{3}+{cos}\left(\mathrm{2}{x}\right)}\right)\right) \\ $$$$\left.\mathrm{2}\right)\:{we}\:{have}\:{A}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\:{C}_{{n}} ^{{k}} \:{cos}^{{k}} \left(\frac{\pi}{{n}}\right)\:{cos}\left(\frac{{k}\pi}{{n}}\right) \\ $$$$={S}_{{n}} \left(\frac{\pi}{{n}}\right)\:=\left(\frac{\mathrm{3}+{cos}\left(\frac{\mathrm{2}\pi}{{n}}\right)}{\mathrm{2}}\right)^{\frac{{n}}{\mathrm{2}}} \:{cos}\left({narctan}\left(\frac{{sin}\left(\frac{\mathrm{2}\pi}{{n}}\right)}{\mathrm{3}+{cos}\left(\frac{\mathrm{2}\pi}{{n}}\right)}\right)\right). \\ $$

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