Prove that cos(θ+30°)cos(θ+60°) = (1/4)√3 - (1/2)sin 2θ.

Mark scheme

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

1. [B1] State (cosθ cos 30 - sinθ sin 30)(cosθ cos 60 - sinθ sin 60)
2. [M1] Expand and use correct exact values
3. [A1] Obtain (1/4)√3(cos²θ+sin²θ) - sinθ cosθ or similarly simplified equivalent
4. [A1] Conclude (1/4)√3 - (1/2)sin 2θ AG – necessary detail needed.
5. [*M1] Use identity to obtain value for sin 4α
6. [A1] Obtain sin 4α = (5√3-4)/10 or 0.466...
7. [DM1] Show correct process to obtain one value of α
8. [A1] Obtain 6.9 and 38.1
9. [B1] Substitute θ = -10 to obtain cos 20cos 50 = (1/4)√3 - (1/2)sin(-20)
10. [B1] Substitute θ = 10 to obtain cos 40 cos 70 = (1/4)√3 - (1/2)sin(20)
11. [B1] Add and confirm (1/2)√3 with clear indication that sin(-20) = -sin 20