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Question Number 50375 by prof Abdo imad last updated on 16/Dec/18

let f(x)=(1/(cosx)) prove that f^()n)) (x)=((p_n (sinx))/(cos^(n+1) x)) with p_n is apolynom 2) calculate p_1 ,p_2 and p_3 3) detdrmine p_n (1).

$${let}\:{f}\left({x}\right)=\frac{\mathrm{1}}{{cosx}}\:{prove}\:{that}\:{f}^{\left.\right)\left.{n}\right)} \left({x}\right)=\frac{{p}_{{n}} \left({sinx}\right)}{{cos}^{{n}+\mathrm{1}} {x}} \\ $$$${with}\:{p}_{{n}} \:{is}\:{apolynom} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:{p}_{\mathrm{1}} ,{p}_{\mathrm{2}} \:{and}\:{p}_{\mathrm{3}} \\ $$$$\left.\mathrm{3}\right)\:{detdrmine}\:{p}_{{n}} \left(\mathrm{1}\right). \\ $$

Answered by kaivan.ahmadi last updated on 10/Jan/19

f′(x)=((sinx)/(cos^2 x))⇒p_1 (x)=x f^(′′) (x)=((cos^3 x+2sin^2 xcosx)/(cos^4 x))=((cos^2 x+2sin^2 x)/(cos^3 x))= ((1+sin^2 x)/(cos^3 x))⇒p_2 (x)=x^2 +1 f^((3)) (x)=((2sinxcos^4 x+3sinxcos^2 x(1+sin^2 x))/(cos^6 x))= ((2sinxcos^4 x+3sinxcos^2 x+3sin^3 xcos^2 x)/(cos^6 x))= ((2sinxcos^2 x+3sinx+3sin^3 x)/(cos^4 x))= ((5sinx+sin^3 x)/(cos^4 x))⇒p_3 (x)=x^3 +5x ⋮ f^((4)) (x)=((5cos^5 x+3sin^2 xcos^5 x+20sin^2 xcos^3 x+4sin^4 xcos^3 x)/(cos^8 x))= ((5cos^2 x+3sin^2 xcos^2 x+20sin^2 x+4sin^4 x)/(cos^5 x))= ((5−5sin^2 x+3sin^2 x−3sin^4 x+20sin^2 x+4sin^4 x)/(cos^5 x))= ((5+18sin^2 x+sin^4 x)/(cos^5 x))⇒p_4 (x)=x^4 +18x^2 +5 ⋮ p_1 (1)=1=1! p_2 (1)=2=2! p_3 (1)=6=3! p_4 (1)=24=4! ⋮ p_n (1)=n!

$$\mathrm{f}'\left(\mathrm{x}\right)=\frac{\mathrm{sinx}}{\mathrm{cos}^{\mathrm{2}} \mathrm{x}}\Rightarrow\mathrm{p}_{\mathrm{1}} \left(\mathrm{x}\right)=\mathrm{x} \\ $$$$\mathrm{f}^{''} \left(\mathrm{x}\right)=\frac{\mathrm{cos}^{\mathrm{3}} \mathrm{x}+\mathrm{2sin}^{\mathrm{2}} \mathrm{xcosx}}{\mathrm{cos}^{\mathrm{4}} \mathrm{x}}=\frac{\mathrm{cos}^{\mathrm{2}} \mathrm{x}+\mathrm{2sin}^{\mathrm{2}} \mathrm{x}}{\mathrm{cos}^{\mathrm{3}} \mathrm{x}}= \\ $$$$\frac{\mathrm{1}+\mathrm{sin}^{\mathrm{2}} \mathrm{x}}{\mathrm{cos}^{\mathrm{3}} \mathrm{x}}\Rightarrow\mathrm{p}_{\mathrm{2}} \left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{2}} +\mathrm{1} \\ $$$$\mathrm{f}^{\left(\mathrm{3}\right)} \left(\mathrm{x}\right)=\frac{\mathrm{2sinxcos}^{\mathrm{4}} \mathrm{x}+\mathrm{3sinxcos}^{\mathrm{2}} \mathrm{x}\left(\mathrm{1}+\mathrm{sin}^{\mathrm{2}} \mathrm{x}\right)}{\mathrm{cos}^{\mathrm{6}} \mathrm{x}}= \\ $$$$\frac{\mathrm{2sinxcos}^{\mathrm{4}} \mathrm{x}+\mathrm{3sinxcos}^{\mathrm{2}} \mathrm{x}+\mathrm{3sin}^{\mathrm{3}} \mathrm{xcos}^{\mathrm{2}} \mathrm{x}}{\mathrm{cos}^{\mathrm{6}} \mathrm{x}}= \\ $$$$\frac{\mathrm{2sinxcos}^{\mathrm{2}} \mathrm{x}+\mathrm{3sinx}+\mathrm{3sin}^{\mathrm{3}} \mathrm{x}}{\mathrm{cos}^{\mathrm{4}} \mathrm{x}}= \\ $$$$\frac{\mathrm{5sinx}+\mathrm{sin}^{\mathrm{3}} \mathrm{x}}{\mathrm{cos}^{\mathrm{4}} \mathrm{x}}\Rightarrow\mathrm{p}_{\mathrm{3}} \left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{3}} +\mathrm{5x} \\ $$$$\vdots \\ $$$$\mathrm{f}^{\left(\mathrm{4}\right)} \left(\mathrm{x}\right)=\frac{\mathrm{5cos}^{\mathrm{5}} \mathrm{x}+\mathrm{3sin}^{\mathrm{2}} \mathrm{xcos}^{\mathrm{5}} \mathrm{x}+\mathrm{20sin}^{\mathrm{2}} \mathrm{xcos}^{\mathrm{3}} \mathrm{x}+\mathrm{4sin}^{\mathrm{4}} \mathrm{xcos}^{\mathrm{3}} \mathrm{x}}{\mathrm{cos}^{\mathrm{8}} \mathrm{x}}= \\ $$$$\frac{\mathrm{5cos}^{\mathrm{2}} \mathrm{x}+\mathrm{3sin}^{\mathrm{2}} \mathrm{xcos}^{\mathrm{2}} \mathrm{x}+\mathrm{20sin}^{\mathrm{2}} \mathrm{x}+\mathrm{4sin}^{\mathrm{4}} \mathrm{x}}{\mathrm{cos}^{\mathrm{5}} \mathrm{x}}= \\ $$$$\frac{\mathrm{5}−\mathrm{5sin}^{\mathrm{2}} \mathrm{x}+\mathrm{3sin}^{\mathrm{2}} \mathrm{x}−\mathrm{3sin}^{\mathrm{4}} \mathrm{x}+\mathrm{20sin}^{\mathrm{2}} \mathrm{x}+\mathrm{4sin}^{\mathrm{4}} \mathrm{x}}{\mathrm{cos}^{\mathrm{5}} \mathrm{x}}= \\ $$$$\frac{\mathrm{5}+\mathrm{18sin}^{\mathrm{2}} \mathrm{x}+\mathrm{sin}^{\mathrm{4}} \mathrm{x}}{\mathrm{cos}^{\mathrm{5}} \mathrm{x}}\Rightarrow\mathrm{p}_{\mathrm{4}} \left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{4}} +\mathrm{18x}^{\mathrm{2}} +\mathrm{5} \\ $$$$\vdots \\ $$$$\mathrm{p}_{\mathrm{1}} \left(\mathrm{1}\right)=\mathrm{1}=\mathrm{1}! \\ $$$$\mathrm{p}_{\mathrm{2}} \left(\mathrm{1}\right)=\mathrm{2}=\mathrm{2}! \\ $$$$\mathrm{p}_{\mathrm{3}} \left(\mathrm{1}\right)=\mathrm{6}=\mathrm{3}! \\ $$$$\mathrm{p}_{\mathrm{4}} \left(\mathrm{1}\right)=\mathrm{24}=\mathrm{4}! \\ $$$$\vdots \\ $$$$\mathrm{p}_{\mathrm{n}} \left(\mathrm{1}\right)=\mathrm{n}! \\ $$

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