Eliminating A from the equation XcosA = Ycos(A + 2p/3) = Zcos(A + 4p/3).

where p = pi
Establish that xy + yz + zx = 0

First, use XcosA = Ycos(A+2p/3) to find tanA by expanding cos(A+2p/3).

Do the same considering XcosA = Zcos(A+4p/3)and then equate the two.

This way A is eliminated and you will get the desired result.

Thanks and Regards,
Deepak Kumar
AskIITians Faculty,
B. Tech, IIT Delhi.