Prove that cos(θ+30°)cos(θ+60°) = ¼√3 - ½sin 2θ.

Mark scheme

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

1. [4] State (cosθcos 30 - sinθsin 30)(cosθcos 60 – sinθsin 60) B1
2. Expand and use correct exact values M1
3. Obtain ¼√3(cos²θ+sin²θ) - ½sinθcosθ or similarly simplified equivalent A1
4. Conclude ¼√3 - ½sin 2θ A1 AG – necessary detail needed.
5. [4] Use identity to obtain value for sin4α *M1
6. Obtain sin4α=¼√3-3 or 0.466... A1
7. Show correct process to obtain one value of α DM1
8. Obtain 6.9 and 38.1 A1 Or greater accuracy; and no others between 0° and 90°.
9. [3] Substitute θ = -10 to obtain cos 20cos50=¼√3-½sin(-20) B1
10. Substitute θ = 10 to obtain cos 40cos70=¼√3-½sin 20 B1
11. Add and confirm ½√3 with clear indication that sin(-20) = -sin 20 B1 AG – necessary detail needed.
12. Alternative solution for Question 7(c)
13. Rewrite as sin 70cos 50+ sin 50cos 70 or cos 20sin 40+ cos 40sin 20 or sin 70 sin 40+ cos 70 cos 40 B1
14. Obtain sin120 or sin 60 or cos30 B1
15. Confirm ½√3 B1 AG – necessary detail needed.