If tan-1x+tan-1y+tan-1z=pie, prove x+y+z=xyz - Maths - Inverse Trigonometric Functions

Question

If tan-1x+tan-1y+tan-1z=pie,
prove x+y+z=xyz

Ajith Kumar · Answer

TAN-1X + TAN-1Y + TAN-1Z = PIE
TAN-1 ( X + Y / 1- XY) + TAN-1Z = PIE { TAN-1X + TAN-1Y = TAN-1
( X + Y / 1- XY) }
TAN-1 ( X + Y / 1-XY) + Z = PIE
1- Z[(X + Y) /(1-XY)]
(X + Y / 1 - XY) + Z = TAN(PIE)
1- Z[(X + Y) /(1-XY)]
(X + Y / 1 - XY) + Z = 0
1- Z[(X + Y) /(1-XY)]
[(X + Y) / (1 - XY)] + Z = 0
(X + Y) / (1 - XY) = -Z
X + Y = -Z(1 - XY)
X + Y = -Z + XYZ
X + Y + Z = XYZ