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Question-173970




Question Number 173970 by mnjuly1970 last updated on 22/Jul/22
Commented by Tawa11 last updated on 22/Jul/22
Great sir
$$\mathrm{Great}\:\mathrm{sir} \\ $$
Commented by infinityaction last updated on 22/Jul/22
       sin^(12) x + cos^(12) x  = (m/n)  =  p              let y = sin^2 x          y^5  + (1−y)^(5 )  =  ((11)/(36))            if we let    t = (1/2)−y             ((1/2)+t)^5 +((1/2)−t)^5  =  ((11)/(36))           use binomial theorem               (1/(16))(80t^4 +40t^2 +1) = ((11)/(36))                 t^2  = (1/(12)) ⇒t  =  ±(1/(2(√3)))                   ((1/2)−t)^6 +((1/2)+t)^6  =  p         use binomial theorem               p  =  ((13)/(54)) = (m/n)             m+n = 67
$$\:\:\:\:\:\:\:\mathrm{sin}^{\mathrm{12}} {x}\:+\:\mathrm{cos}^{\mathrm{12}} {x}\:\:=\:\frac{{m}}{{n}}\:\:=\:\:{p}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{let}\:{y}\:=\:\mathrm{sin}^{\mathrm{2}} {x} \\ $$$$\:\:\:\:\:\:\:\:{y}^{\mathrm{5}} \:+\:\left(\mathrm{1}−{y}\right)^{\mathrm{5}\:} \:=\:\:\frac{\mathrm{11}}{\mathrm{36}} \\ $$$$\:\:\:\:\:\:\:\:\:\:{if}\:{we}\:{let}\:\:\:\:{t}\:=\:\frac{\mathrm{1}}{\mathrm{2}}−{y} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\left(\frac{\mathrm{1}}{\mathrm{2}}+{t}\right)^{\mathrm{5}} +\left(\frac{\mathrm{1}}{\mathrm{2}}−{t}\right)^{\mathrm{5}} \:=\:\:\frac{\mathrm{11}}{\mathrm{36}} \\ $$$$\:\:\:\:\:\:\:\:\:{use}\:{binomial}\:{theorem} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{16}}\left(\mathrm{80}{t}^{\mathrm{4}} +\mathrm{40}{t}^{\mathrm{2}} +\mathrm{1}\right)\:=\:\frac{\mathrm{11}}{\mathrm{36}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{t}^{\mathrm{2}} \:=\:\frac{\mathrm{1}}{\mathrm{12}}\:\Rightarrow{t}\:\:=\:\:\pm\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{3}}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\frac{\mathrm{1}}{\mathrm{2}}−{t}\right)^{\mathrm{6}} +\left(\frac{\mathrm{1}}{\mathrm{2}}+{t}\right)^{\mathrm{6}} \:=\:\:{p} \\ $$$$\:\:\:\:\:\:\:{use}\:{binomial}\:{theorem} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{p}\:\:=\:\:\frac{\mathrm{13}}{\mathrm{54}}\:=\:\frac{{m}}{{n}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{m}+{n}\:=\:\mathrm{67} \\ $$
Commented by mnjuly1970 last updated on 22/Jul/22
grateful sir
$${grateful}\:{sir}\: \\ $$
Answered by mr W last updated on 22/Jul/22
let p=sin x cos x  (sin^2  x+cos^2  x)^2 =1  sin^4  x+cos^4  x=1−2p^2   (sin^4  x+cos^4  x)(sin^2  x+cos^2  x)=1−2p^2   sin^6  x+cos^6  x+sin^2  x cos^2  x(sin^2  x+cos^2  x)=1−2p^2   sin^6  x+cos^6  x+p^2 =1−2p^2   sin^6  x+cos^6  x=1−3p^2     (sin^4  x+cos^4  x)^2 =(1−2p^2 )^2   sin^8  x+cos^8  x+2p^4 =1−4p^2 +4p^4   sin^8  x+cos^8  x=1−4p^2 +2p^4     (sin^8  x+cos^8  x)(sin^2  x+cos^2  x)=1−4p^2 +2p^4   sin^(10)  x+cos^(10)  x+sin^2  x cos^2  x(sin^6  x+cos^6  x)=1−4p^2 +2p^4   sin^(10)  x+cos^(10)  x+p^2 (1−3p^2 )=1−4p^2 +2p^4   sin^(10)  x+cos^(10)  x=1−5p^2 +5p^4   ((11)/(36))=1−5p^2 +5p^4   ⇒p^4 −p^2 +(5/(36))=0  ⇒p^2 =(1/2)(1±(2/3))=(5/6)>(1/4) rejected or (1/6) ✓    (sin^(10)  x+cos^(10)  x)(sin^2  x+cos^2  x)=((11)/(36))  sin^(12)  x+cos^(12)  x+sin^2  x cos^2  x(sin^8  x+cos^8  x)=((11)/(36))  sin^(12)  x+cos^(12)  x+p^2 (1−4p^2 +2p^4 )=((11)/(36))  sin^(12)  x+cos^(12)  x=((11)/(36))−p^2 (1−4p^2 +2p^4 )  sin^(12)  x+cos^(12)  x=((11)/(36))−p^2 (1−4p^2 +2p^2 −((10)/(36)))  sin^(12)  x+cos^(12)  x=((11)/(36))−(((26p^2 )/(36))−2p^4 )  sin^(12)  x+cos^(12)  x=((11)/(36))−(((26p^2 )/(36))−2p^2 +((10)/(36)))  sin^(12)  x+cos^(12)  x=((1+46p^2 )/(36))  sin^(12)  x+cos^(12)  x=((1+46×(1/6))/(36))=((13)/(54)) ✓  ⇒m+n=13+54=67  but generally m+n=67k, k∈R ∧ k≠0.
$${let}\:{p}=\mathrm{sin}\:{x}\:\mathrm{cos}\:{x} \\ $$$$\left(\mathrm{sin}^{\mathrm{2}} \:{x}+\mathrm{cos}^{\mathrm{2}} \:{x}\right)^{\mathrm{2}} =\mathrm{1} \\ $$$$\mathrm{sin}^{\mathrm{4}} \:{x}+\mathrm{cos}^{\mathrm{4}} \:{x}=\mathrm{1}−\mathrm{2}{p}^{\mathrm{2}} \\ $$$$\left(\mathrm{sin}^{\mathrm{4}} \:{x}+\mathrm{cos}^{\mathrm{4}} \:{x}\right)\left(\mathrm{sin}^{\mathrm{2}} \:{x}+\mathrm{cos}^{\mathrm{2}} \:{x}\right)=\mathrm{1}−\mathrm{2}{p}^{\mathrm{2}} \\ $$$$\mathrm{sin}^{\mathrm{6}} \:{x}+\mathrm{cos}^{\mathrm{6}} \:{x}+\mathrm{sin}^{\mathrm{2}} \:{x}\:\mathrm{cos}^{\mathrm{2}} \:{x}\left(\mathrm{sin}^{\mathrm{2}} \:{x}+\mathrm{cos}^{\mathrm{2}} \:{x}\right)=\mathrm{1}−\mathrm{2}{p}^{\mathrm{2}} \\ $$$$\mathrm{sin}^{\mathrm{6}} \:{x}+\mathrm{cos}^{\mathrm{6}} \:{x}+{p}^{\mathrm{2}} =\mathrm{1}−\mathrm{2}{p}^{\mathrm{2}} \\ $$$$\mathrm{sin}^{\mathrm{6}} \:{x}+\mathrm{cos}^{\mathrm{6}} \:{x}=\mathrm{1}−\mathrm{3}{p}^{\mathrm{2}} \\ $$$$ \\ $$$$\left(\mathrm{sin}^{\mathrm{4}} \:{x}+\mathrm{cos}^{\mathrm{4}} \:{x}\right)^{\mathrm{2}} =\left(\mathrm{1}−\mathrm{2}{p}^{\mathrm{2}} \right)^{\mathrm{2}} \\ $$$$\mathrm{sin}^{\mathrm{8}} \:{x}+\mathrm{cos}^{\mathrm{8}} \:{x}+\mathrm{2}{p}^{\mathrm{4}} =\mathrm{1}−\mathrm{4}{p}^{\mathrm{2}} +\mathrm{4}{p}^{\mathrm{4}} \\ $$$$\mathrm{sin}^{\mathrm{8}} \:{x}+\mathrm{cos}^{\mathrm{8}} \:{x}=\mathrm{1}−\mathrm{4}{p}^{\mathrm{2}} +\mathrm{2}{p}^{\mathrm{4}} \\ $$$$ \\ $$$$\left(\mathrm{sin}^{\mathrm{8}} \:{x}+\mathrm{cos}^{\mathrm{8}} \:{x}\right)\left(\mathrm{sin}^{\mathrm{2}} \:{x}+\mathrm{cos}^{\mathrm{2}} \:{x}\right)=\mathrm{1}−\mathrm{4}{p}^{\mathrm{2}} +\mathrm{2}{p}^{\mathrm{4}} \\ $$$$\mathrm{sin}^{\mathrm{10}} \:{x}+\mathrm{cos}^{\mathrm{10}} \:{x}+\mathrm{sin}^{\mathrm{2}} \:{x}\:\mathrm{cos}^{\mathrm{2}} \:{x}\left(\mathrm{sin}^{\mathrm{6}} \:{x}+\mathrm{cos}^{\mathrm{6}} \:{x}\right)=\mathrm{1}−\mathrm{4}{p}^{\mathrm{2}} +\mathrm{2}{p}^{\mathrm{4}} \\ $$$$\mathrm{sin}^{\mathrm{10}} \:{x}+\mathrm{cos}^{\mathrm{10}} \:{x}+{p}^{\mathrm{2}} \left(\mathrm{1}−\mathrm{3}{p}^{\mathrm{2}} \right)=\mathrm{1}−\mathrm{4}{p}^{\mathrm{2}} +\mathrm{2}{p}^{\mathrm{4}} \\ $$$$\mathrm{sin}^{\mathrm{10}} \:{x}+\mathrm{cos}^{\mathrm{10}} \:{x}=\mathrm{1}−\mathrm{5}{p}^{\mathrm{2}} +\mathrm{5}{p}^{\mathrm{4}} \\ $$$$\frac{\mathrm{11}}{\mathrm{36}}=\mathrm{1}−\mathrm{5}{p}^{\mathrm{2}} +\mathrm{5}{p}^{\mathrm{4}} \\ $$$$\Rightarrow{p}^{\mathrm{4}} −{p}^{\mathrm{2}} +\frac{\mathrm{5}}{\mathrm{36}}=\mathrm{0} \\ $$$$\Rightarrow{p}^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}\pm\frac{\mathrm{2}}{\mathrm{3}}\right)=\frac{\mathrm{5}}{\mathrm{6}}>\frac{\mathrm{1}}{\mathrm{4}}\:{rejected}\:{or}\:\frac{\mathrm{1}}{\mathrm{6}}\:\checkmark \\ $$$$ \\ $$$$\left(\mathrm{sin}^{\mathrm{10}} \:{x}+\mathrm{cos}^{\mathrm{10}} \:{x}\right)\left(\mathrm{sin}^{\mathrm{2}} \:{x}+\mathrm{cos}^{\mathrm{2}} \:{x}\right)=\frac{\mathrm{11}}{\mathrm{36}} \\ $$$$\mathrm{sin}^{\mathrm{12}} \:{x}+\mathrm{cos}^{\mathrm{12}} \:{x}+\mathrm{sin}^{\mathrm{2}} \:{x}\:\mathrm{cos}^{\mathrm{2}} \:{x}\left(\mathrm{sin}^{\mathrm{8}} \:{x}+\mathrm{cos}^{\mathrm{8}} \:{x}\right)=\frac{\mathrm{11}}{\mathrm{36}} \\ $$$$\mathrm{sin}^{\mathrm{12}} \:{x}+\mathrm{cos}^{\mathrm{12}} \:{x}+{p}^{\mathrm{2}} \left(\mathrm{1}−\mathrm{4}{p}^{\mathrm{2}} +\mathrm{2}{p}^{\mathrm{4}} \right)=\frac{\mathrm{11}}{\mathrm{36}} \\ $$$$\mathrm{sin}^{\mathrm{12}} \:{x}+\mathrm{cos}^{\mathrm{12}} \:{x}=\frac{\mathrm{11}}{\mathrm{36}}−{p}^{\mathrm{2}} \left(\mathrm{1}−\mathrm{4}{p}^{\mathrm{2}} +\mathrm{2}{p}^{\mathrm{4}} \right) \\ $$$$\mathrm{sin}^{\mathrm{12}} \:{x}+\mathrm{cos}^{\mathrm{12}} \:{x}=\frac{\mathrm{11}}{\mathrm{36}}−{p}^{\mathrm{2}} \left(\mathrm{1}−\mathrm{4}{p}^{\mathrm{2}} +\mathrm{2}{p}^{\mathrm{2}} −\frac{\mathrm{10}}{\mathrm{36}}\right) \\ $$$$\mathrm{sin}^{\mathrm{12}} \:{x}+\mathrm{cos}^{\mathrm{12}} \:{x}=\frac{\mathrm{11}}{\mathrm{36}}−\left(\frac{\mathrm{26}{p}^{\mathrm{2}} }{\mathrm{36}}−\mathrm{2}{p}^{\mathrm{4}} \right) \\ $$$$\mathrm{sin}^{\mathrm{12}} \:{x}+\mathrm{cos}^{\mathrm{12}} \:{x}=\frac{\mathrm{11}}{\mathrm{36}}−\left(\frac{\mathrm{26}{p}^{\mathrm{2}} }{\mathrm{36}}−\mathrm{2}{p}^{\mathrm{2}} +\frac{\mathrm{10}}{\mathrm{36}}\right) \\ $$$$\mathrm{sin}^{\mathrm{12}} \:{x}+\mathrm{cos}^{\mathrm{12}} \:{x}=\frac{\mathrm{1}+\mathrm{46}{p}^{\mathrm{2}} }{\mathrm{36}} \\ $$$$\mathrm{sin}^{\mathrm{12}} \:{x}+\mathrm{cos}^{\mathrm{12}} \:{x}=\frac{\mathrm{1}+\mathrm{46}×\frac{\mathrm{1}}{\mathrm{6}}}{\mathrm{36}}=\frac{\mathrm{13}}{\mathrm{54}}\:\checkmark \\ $$$$\Rightarrow{m}+{n}=\mathrm{13}+\mathrm{54}=\mathrm{67} \\ $$$${but}\:{generally}\:{m}+{n}=\mathrm{67}{k},\:{k}\in{R}\:\wedge\:{k}\neq\mathrm{0}. \\ $$
Commented by Tawa11 last updated on 22/Jul/22
Great sir
$$\mathrm{Great}\:\mathrm{sir} \\ $$
Commented by mnjuly1970 last updated on 22/Jul/22
   excellent sirW
$$\:\:\:{excellent}\:{sirW} \\ $$
Commented by peter frank last updated on 24/Jul/22
thank you
$$\mathrm{thank}\:\mathrm{you} \\ $$

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