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

Explanation

LHS `=cos alpha+cos beta+cos gamma+cos(alpha+beta+gamma)`
`=(cos alpha+cos beta)+(cos gamma+cos(alpha+beta+gamma)`
`=2 cos((alpha+beta)/(2))cos((alpha+beta)/(2))`
`+2cos((alpha+beta+gamma+gamma)/(2))cos((alpha+beta+gamma-gamma)/(2))`
`=2cos((alpha+beta)/(2))cos((alpha-beta)/(2))`
`+2cos((alpha+beta)/(2))cos((alpha+beta+2gamma)/(2))`
`=2cos((alpha+beta)/(2)){cos((alpha-beta)/(2))+cos((alpha+beta+2gamma)/(2))}`
`=2cos((alpha+beta)/(2)){2cos(((alpha-beta)/(2)+(alpha+beta+2gamma)/(2))/(2))`
`cos(((alpha+beta+2gamma)/(2)-(alpha-beta)/(2))/(2))}`
`=2cos((alpha+beta)/(2){2cos((alpha+gamma)/(2))cos((beta+gamma)/(2))}`
`=4cos((alpha+beta)/(2))cos((beta+gamma)/(2))cos((gamma+alpha)/(2))`=RHS.