If-x-1-x-2cos-y-1-y-2cos-z-1-z-2cos-Show-that-xyz-1-xyz-2cos-

Question Number 51897 by Tawa1 last updated on 31/Dec/18

If x + (1/x) = 2cosθ , y + (1/y) = 2cosφ , z + (1/z) = 2cosψ Show that xyz + (1/(xyz)) = 2cos(θ + φ + ψ)

$$\mathrm{If}\:\:\:\:\mathrm{x}\:+\:\frac{\mathrm{1}}{\mathrm{x}}\:\:=\:\:\mathrm{2cos}\theta\:,\:\:\:\:\:\:\mathrm{y}\:+\:\frac{\mathrm{1}}{\mathrm{y}}\:\:=\:\:\mathrm{2cos}\phi\:,\:\:\:\:\:\:\:\:\:\mathrm{z}\:+\:\frac{\mathrm{1}}{\mathrm{z}}\:\:=\:\:\mathrm{2cos}\psi \\ $$$$\mathrm{Show}\:\mathrm{that}\:\:\:\:\:\:\:\:\:\:\mathrm{xyz}\:+\:\frac{\mathrm{1}}{\mathrm{xyz}}\:\:=\:\:\mathrm{2cos}\left(\theta\:+\:\phi\:+\:\psi\right) \\ $$

Answered by behi83417@gmail.com last updated on 31/Dec/18

cosθ=((e^(iθ) +e^(−iθ) )/2)=(1/2)(e^(iθ) +(1/e^(iθ) ))=(1/2)(x+(1/x)) we can put: x=e^(iθ) ,y=e^(i∅) ,z=e^(iψ) ⇒xyz+(1/(xyz))=e^(iθ+iφ+iψ) +(1/e^(iθ+iφ+iψ) )= =e^(i(θ+φ+ψ)) +(1/e^(i(θ+φ+ψ)) )=2cos(θ+φ+ψ) .

$${cos}\theta=\frac{{e}^{{i}\theta} +{e}^{−{i}\theta} }{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}}\left({e}^{{i}\theta} +\frac{\mathrm{1}}{{e}^{{i}\theta} }\right)=\frac{\mathrm{1}}{\mathrm{2}}\left({x}+\frac{\mathrm{1}}{{x}}\right) \\ $$$${we}\:{can}\:{put}:\:\:\:{x}={e}^{{i}\theta} ,{y}={e}^{{i}\emptyset} ,{z}={e}^{{i}\psi} \\ $$$$\Rightarrow{xyz}+\frac{\mathrm{1}}{{xyz}}={e}^{{i}\theta+{i}\phi+{i}\psi} +\frac{\mathrm{1}}{{e}^{{i}\theta+{i}\phi+{i}\psi} }= \\ $$$$={e}^{{i}\left(\theta+\phi+\psi\right)} +\frac{\mathrm{1}}{{e}^{{i}\left(\theta+\phi+\psi\right)} }=\mathrm{2}{cos}\left(\theta+\phi+\psi\right)\:. \\ $$

Commented by Tawa1 last updated on 01/Jan/19

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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If-x-1-x-2cos-y-1-y-2cos-z-1-z-2cos-Show-that-xyz-1-xyz-2cos-

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