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

Explanation

Given,`cos alpha+cos beta=0=sin alpha+sin beta`


Squaring on both sides gives

`=>(cos alpha+cos beta)^2=(sin alpha+sin beta)^2`

`=>cos^2 alpha+cos^2 beta+2 cos alpha cos beta=sin^2 alpha+sin^2beta+2 sin alphasin beta`

`=>cos2alpha+cos2beta+2cosalphacosbeta−sin2alpha−sin2beta−2sinalphasinbeta=0 `


`=>(cos^2alpha−sin^2alpha)+(cos^2beta−sin^2beta)=2(−sinalphasinbeta+cosalphacosbeta) `
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