यदि `cos (beta-gamma)+cos(gamma-alpha)+cos(alpha-beta)= - (3)/(2)`
साबित करें कि `cos alpha+cosbeta+cos gamma =sinalpha+sinbeta+sin gamma = 0`

Explanation

प्रश्न से, `cos(beta-gamma)+cos(gamma-alpha)+cos(alpha-beta)= - (3)/(2)`
`implies 3+2cos (beta-gamma) + 2cos ( gamma-alpha)+2cos ( alpha-beta)=0`
`implies 3+2(cos beta cos gamma +sin beta sin gamma) + 2(cos gamma cos alpha +sin gamma sin alpha) +2(cos gamma cos beta +sin alpha sin beta)=0`
`implies (cos^(2)alpha+sin^(2)alpha) + (cos^(2)beta+sin^(2)beta)+(cos^(2)gamma+sin^(2)gamma)+2(cosbetacosgamma+sinbeta sin gamma)+2(cos gamma cos alpha + sin gamma sin alpha)+2(cosalphacos beta +sin alpha sin beta )=0`
`implies (cos^(2)alpha+cos^(2)beta+cos^(2)gamma+2cosalphacosbeta+2cosbetacosgamma +2cos gamma cos gamma) + (sin^(2)alpha+sin^(2)beta+sin^(2)gamma+2sinalphasinbeta+2sin beta sin gamma + 2sin gamma sin gamma sin alpha ) = 0`
`implies ( cos alpha +cos beta+cos gamma)^(2) + ( sin alpha+sin beta+sin gamma)^(2) = 0 `
जो तभी संभव है जब
`cos alpha +cos beta+cos gamma = 0` और `sin alpha+sin beta +sin gamma=0`