Correct answer: True (identity proven)
Since arccot(1/y) = arctan(y), the relation becomes arctan(x) + arctan(y) + 2*arctan(z) = pi, i.e. arctan(x) + arctan(y) = pi - 2*arctan(z). Take tangent of both sides. LHS: tan(arctan x + arctan y) = (x + y)/(1 - x*y). RHS: tan(pi - 2*arctan z) = -tan(2*arctan z) = -2z/(1 - z²). Equate: (x + y)/(1 - x*y) = -2z/(1 - z²). Cross-multiply: (x + y)(1 - z²) = -2z(1 - x*y), i.e. (x + y) - (x + y)z² = -2z + 2xyz. Rearranging: x + y + 2z = (x + y)z² + 2xyz = x*z² + y*z² + 2*x*y*z. Proven.