Question

How to solve this identity?

Prove that

`cosx + cosy + cosz + cos(x+ y+ z)=4cos((x+y)/2) * cos((y+z)/2) * cos((z+x)/2)`

Thank you!

Answer 1

`cosx + cosy + cosz + cos(x+ y+ z)`
`=(cosx+cosy)+{cosz+cos(x+y+z)}`
`=2cdot cos((x+y)/2)cos((x-y)/2)+2cdot cos((x+y+2z)/2)cdot cos((x+y+z-z)/2)`
`=2cdot cos((x+y)/2)cos((x-y)/2)+2cdot cos((x+y+2z)/2)cdot cos((x+y)/2)`
`=2cdot cos((x+y)/2){cos((x-y)/2)+cos((x+y+2z)/2)}`
`=2cdot cos((x+y)/2){2cdotcos(((x-y)/2+(x+y+2z)/2)/2)cdotcos((-(x-y)/2+(x+y+z)/2)/2)}`
`=2cdot cos((x+y)/2){2cdotcos(((x-y)/2+(x+y+2z)/2)/2)cdotcos((-(x-y)/2+(x+y+z)/2)/2)}`
`=2cdot cos((x+y)/2){2cdotcos(((2(x+z))/2)/2)cdotcos(((2(y+z))/2)/2)}`
`=2cdotcos((x+y)/2)cdotcos((y+z)/2)cdotcos((x+z)/2)`

Hope it helps...
Thank you...