How do you prove: $\frac{sin (α - β)}{sin α sin β} + \frac{sin (β - γ)}{sin β sin γ} + \frac{sin (γ - α)}{sin γ sin α} = 0$ $sin(alpha-beta)/(sinalphasinbeta) + sin(beta - gamma)/(sinbetasingamma) + sin(gamma-alpha)/(singammasinalpha) = 0$ ?

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How do you prove: $\frac{sin (α - β)}{sin α sin β} + \frac{sin (β - γ)}{sin β sin γ} + \frac{sin (γ - α)}{sin γ sin α} = 0$ $sin(alpha-beta)/(sinalphasinbeta) + sin(beta - gamma)/(sinbetasingamma) + sin(gamma-alpha)/(singammasinalpha) = 0$ ?

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Answer 1 · 2018-04-19T14:01:32

To prove

$\frac{sin (α - β)}{sin α sin β} + \frac{sin (β - γ)}{sin β sin γ} + \frac{sin (γ - α)}{sin γ sin α} = 0$ $sin(alpha-beta)/(sinalphasinbeta) + sin(beta - gamma)/(sinbetasingamma) + sin(gamma-alpha)/(singammasinalpha) = 0$

1st part

$\frac{sin (α - β)}{sin α sin β}$ $sin(alpha-beta)/(sinalphasinbeta)$

$= \frac{sin α cos β - cos α sin β}{sin α sin β}$ $=(sinalphacosbeta-cosalphasinbeta)/(sinalphasinbeta)$

$= \frac{sin α cos β}{sin α sin β} - \frac{cos α sin β}{sin α sin β}$ $=(sinalphacosbeta)/(sinalphasinbeta)-(cosalphasinbeta)/(sinalphasinbeta)$

$= cot β - cot α$ $=cotbeta-cotalpha$

Similarly

2nd part

$= \frac{sin (β - γ)}{sin β sin γ} = cot γ - cot β$ $=sin(beta - gamma)/(sinbetasingamma)=cotgamma-cotbeta$

And 3rd part

$= \frac{sin (γ - α)}{sin γ sin α} = cot α - cot γ$ $=sin(gamma-alpha)/(singammasinalpha)=cotalpha-cotgamma$

So whole expression

$= cot β - cot α + cot γ - cot β + cot α - cot γ = 0$ $=cotbeta-cotalpha+cotgamma-cotbeta+cotalpha-cotgamma=0$

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How do you prove: $\frac{sin (α - β)}{sin α sin β} + \frac{sin (β - γ)}{sin β sin γ} + \frac{sin (γ - α)}{sin γ sin α} = 0$ $sin(alpha-beta)/(sinalphasinbeta) + sin(beta - gamma)/(sinbetasingamma) + sin(gamma-alpha)/(singammasinalpha) = 0$ ?

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How do you prove: sin(α−β)sinαsinβ+sin(β−γ)sinβsinγ+sin(γ−α)sinγsinα=0sin(alpha-beta)/(sinalphasinbeta) + sin(beta - gamma)/(sinbetasingamma) + sin(gamma-alpha)/(singammasinalpha) = 0?

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How do you prove: $\frac{sin (α - β)}{sin α sin β} + \frac{sin (β - γ)}{sin β sin γ} + \frac{sin (γ - α)}{sin γ sin α} = 0$ $sin(alpha-beta)/(sinalphasinbeta) + sin(beta - gamma)/(sinbetasingamma) + sin(gamma-alpha)/(singammasinalpha) = 0$ ?