Prove the identity sin 4x = 4 sinx(2 cos³x-cosx).

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.

1. [1] (a) Use correct double angle formula to expand sin4x (e.g., sin4x = 2sin2xcos2x).
2. [1] (a) Use correct double angle formulae to obtain an expression in sin x and cos x. E.g., 2(2sinxcosx)(2cos²x−1) = 4sinxcosx(2cos²x−1).
3. [1] (a) Obtain sin 4x = 4sinx(2cos³x−cosx) from fully correct working.
4. [1] (b) Use the identity to obtain ∫ 8sinxcos⁶x dx - ∫ 4sinxcos⁴x dx.
5. [1] (b) Obtain -8/7 cos⁷x + 4/5 cos⁵x (or r cos⁷x + s cos⁵x, or equivalent form using substitution u=cosx, -8/7 u⁷ + 4/5 u⁵).
6. [1] (b) Use the correct limits correctly in an expression of the form rcos⁷x+ scos⁵x.
7. [1] (b) Obtain (1/35)(√2+12). Or exact simplified equivalent (e.g. as 2 terms).