Prove that sin 2θ(a cotθ+b tan θ) ≡ a+b+ (a−b) cos 2θ, where a and b are constants.

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2024). The mark or mark type is shown in brackets.

1. [B1] (a) Use correct identity for sin2θ or for cotθ
2. [M1] (a) Express left-hand side in terms of sinθ and cosθ
3. [M1] (a) Attempt to express k₁ cos² θ + k₂ sin² θ in terms of cos 2θ only
4. [A1] (a) Confirm a+b+(a−b)cos2θ [Answer given – necessary detail needed.]
5. [B1] (b) Obtain integrand 8+2cos2θ
6. [B1] (b) Integrate to obtain 8θ+sin2θ
7. [B1] (b) Apply limits correctly to obtain π/3 + ½√3 - (π/6 + ½√3)
8. [B1] (b) Obtain π/6 [Or exact equivalent.]
9. [B1] (c) Use identity to obtain 5cos²α = -2
10. [B1] (c) Obtain α = 2.97 [Or greater accuracy.]
11. [B1] (c) Obtain α = -2.97 [Or greater accuracy; and no others between -π and π.]