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Given that arctan(x) + arccot(1/y) + 2*arctan(z) = pi, prove that x + y + 2z = x*z² + y*z² + 2*x*y*z.

1. True (identity proven)
2. False
3. True only if z = 0
4. True only if x = y

Correct answer: True (identity proven)

Solution

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.

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