2010/028) If x + y + z = xyz

Prove that x(1-y^2)(1-z^2) + y(1-z^2)(1-x^2) + z(1-x^2)(1-y^2) = 4xyz?

proof

If x + y + z = xyz,

putting x = tan A, y = tan B, z = tan C,

tan A + tan B + tan C = tan A tan B tan C, i.e., S₁ = S₃

∴ tan ( A+B+C ) = ( S₁ - S₃ ) / ( 1 - S₂ ) = 0

∴ A+B+C = nπ for integral n.

2A + 2B +2C = 2nπ

∴ tan ( 2A+ 2B+ 2C ) = tan 2nπ = 0

∴ tan 2A + tan 2B + tan 2C = tan 2A tan 2B tan 2C

∴ 2x / (1-x²) + 2y / (1-y²) + 2z / (1-z²) = [ 2x / (1-x²) ]•[ 2y / (1-y²) ]•[ 2z / (1-z²) ]

∴ x(1-y²)(1-z²) + y(1-z²)(1-x²) + z(1-x²)(1-y²) = 4xyz ............

Hence proved