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If cosalpha+cosbeta=0=sinalpha+sinbeta, ...

If `cosalpha+cosbeta=0=sinalpha+sinbeta`, then prove that `cos2alpha +cos2beta=-2cos(alpha +beta)`.

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Given that, `" "cosalpha+cosbeta=0=sinalpha+sinbeta`
`rArr" "(cosalpha+cosbeta)^(2)-(sinalpha+sinbeta)^(2)=0`
`rArr" "cos^(2)alpha-sin^(2)beta+2cosalphacosbeta-sin^(2)alpha-sin^(2)beta-2sinalphasinbeta= 0`
`rArr" "cos^(2)alpha-sin^(2)alpha+cos^(2)beta-sin^(2)beta=2(sinalphasinbeta-cosalphacosbeta)` ` rArr" "cos2alpha+cos2beta=-2cos(alpha+beta)" "` Hence proved.
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