How do you prove: #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

#sin(alpha-beta)/(sinalphasinbeta) + sin(beta - gamma)/(sinbetasingamma) + sin(gamma-alpha)/(singammasinalpha) = 0#

1st part

#sin(alpha-beta)/(sinalphasinbeta) #

#=(sinalphacosbeta-cosalphasinbeta)/(sinalphasinbeta) #

#=(sinalphacosbeta)/(sinalphasinbeta)-(cosalphasinbeta)/(sinalphasinbeta) #

#=cotbeta-cotalpha#

Similarly

2nd part

# =sin(beta - gamma)/(sinbetasingamma)=cotgamma-cotbeta#

And 3rd part

#=sin(gamma-alpha)/(singammasinalpha)=cotalpha-cotgamma#

So whole expression

#=cotbeta-cotalpha+cotgamma-cotbeta+cotalpha-cotgamma=0#