Let `alpha,betaa n dgamma` satisfy `0 < alpha < beta < gamma < 2pidot` It `cos(x+alpha)+cos(x+beta)+cos(x+gamma)=0` for all`x in R ,` then find the possible values of `(gamma-alpha)`

Explanation

`cos(x+alpha)+cos(x+beta)+cos(x+gamma) = 0`

`=>cosx(cosalpha+cosbeta+cosgamma) - sinx(sinalpha+sinbeta+singamma) = 0`

Solution of the above equatio is possible only if,

`cosalpha+cosbeta+cosgamma = sinalpha+sinbeta+singamma =0`

`=>cosalpha+cosgamma = - cosbeta->(1)`

`=>sinalpha+singamma = -sinbeta->(2)`

Squaring and adding (1) and (2),

`=>cos^2alpha+cos^2gamma+2cosalphacosgamma+sin^2alpha+sin^2gamma+2sinalphasingamma = cos^2beta+sin^2beta`

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