Prove that:

`cosalpha+cosbeta+cosgamma+cos(alpha+beta+gamma)=4cos(alpha+beta)/2cos(beta+gamma)/2cos(gamma+alpha)/2` .

Explanation

`text { LHS }=cos alpha+cos beta+cos gamma+cos (a+beta+gamma) `


`=(cos a+cos beta)+[cos gamma+cos (a+beta+gamma)] `


`=2 cos (frac{a+beta}{2}) cdot cos (frac{a-beta}{2})+2 cos (frac{gamma+a+beta+gamma}{2}) cdot cos (frac{y-alpha-beta-gamma}{2}) `


`[ cos x+cos y=2 cos (frac{x+y}{2}) cos (frac{x-y}{2})]`


`=2 cos (frac{a+beta}{2}) cdot cos (frac{a-beta}{2})+2 cos (frac{a+beta+2 gamma}{2}) cdot cos (frac{a+beta}{2}) `


`[ cos (-x)=cos x] `


`=2 cos (frac{a+beta}{2})[cos (frac{a-beta}{2})+cos (frac{a+beta+2 gamma}{2})] `


`=2 cos (frac{a+beta}{2})[2 cos (frac{frac{a-beta}{2}+frac{a+beta+2 y}{2}}{2}) cdot cos (frac{frac{a-beta}{2}-frac{a+beta+2 gamma}{2}}{2})] `


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