7 (a) By expressing 3θ as 2θ + θ, prove the identity cos 3θ = 4 cos³ θ – 3 cos θ. (b) Hence solve the equation cos 3θ + cos θ cos 2θ = cos² θ for 0° ≤ θ ≤ 180°.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. *M1 Use correct expansion for cos(2θ+θ)
2. DM1 Use correct double angle formulae to express cos3θ in terms of cosθ and sinθ
3. A1 AG Show sufficient working to confirm cos3θ=4cos³θ–3cosθ
4. *M1 Use the identity and correct double angle formula to obtain an equation in cosθ only. Must come from using all three terms in the given equation. e.g. 4 cos³θ-3 cosθ + cosθ (2 cos²θ− 1) = cos²θ or 6 cos³θ - cos²θ - 4 cosθ = 0 or 6 cos²θ - cosθ - 4 = 0.
5. B1 Obtain θ=90° Allow if cosθ obtained correctly as a factor of their expression (even if there is an error in the quadratic factor). Can follow M0.
6. DM1 Solve a 3-term quadratic in cosθ to obtain a value of θ
7. A1 Obtain one value e.g. 25.3° Accept awrt 25.3°.
8. A1 Obtain a second value e.g. 137.5° and no extras in range Accept awrt 137.5° . Ignore values outside the range. Mark solutions in radians as a misread (0.442, 1.57, 2.40).