Zadanie 9a, Paper 1, listopad 2024, Matematyka AA IB HL

Write the cotangent as cos(θ+π/4)/sin(θ+π/4). Expanding the compound angles gives $(\cos\theta-\sin\theta)/(\sin\theta+\cos\theta)$. Multiply numerator and denominator by cosθ−sinθ. The numerator becomes cos²θ−2sinθcosθ+sin²θ=1−sin2θ, and the denominator cos²θ−sin²θ=cos2θ. The exclusions ensure the required denominators and multiplier are nonzero, establishing the identity.

M1: express cotangent as a sine/cosine ratio or use an equivalent addition identity. M1: expand the compound angles. A1: correct ratio (cosθ−sinθ)/(sinθ+cosθ). M1: multiply by the conjugate expression. A1: correct squared-numerator and difference-of-squares denominator. A1: reduce using Pythagorean/double-angle identities to establish the target. Equivalent tangent/cotangent addition-formula derivations receive the same six marks: M1 attempt the addition formula, A1 correct quotient, M1 conjugate multiplication, A1 correct product quotient, M1 conversion to sine/cosine or equivalent identities, A1 simplification establishing the result.