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!

1 Answer
Sunayan S
Dec 9, 2017

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...

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