Prove that 4 sin x sin(x + π/6) = √3 – √3 cos 2x + sin 2x.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

1. [B1] State 4 sin x (sin x cos π/6 + cos x sin π/6) Allow marks if completed correctly, starting with RHS.
2. [M1] Attempt relevant identities to express at least one term in terms of cos 2x and sin 2x
3. [A1] Confirm √3 – √3 cos 2x + sin 2x with sufficient detail AG
4. [*M1] Integrate to obtain form k₁x + k₂ sin 2x + k₃ cos 2x Where k₁k₂k₃ ≠ 0
5. [A1] Obtain correct √3x – √3/2 sin 2x – 1/2 cos 2x
6. [DM1] Apply limits correctly and attempt simplification, retaining exactness Condone one sign slip.
7. [A1] Obtain 5π/3 + 1 or exact equivalent
8. [M1] Apply identity with x replaced by 2y May be implied later.
9. [M1] Attempt solution of tan 4y = √3
10. [A1] Obtain 4y = π/3 and hence y = π/12