Show that the equation sin 2θ + cos 2θ = 2 sin² θ can be expressed in the form cos² θ + 2 sin θ cos θ – 3 sin² θ = 0.

Mark scheme

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

1. [M1] Use correct double angle formulae e.g. 2 sin θ cos θ + cos² θ – sin² θ = 2sin² θ
2. [A1] Obtain cos² θ + 2 sin θ cos θ – 3 sin² θ = 0 from full and correct working AG Check conclusion is complete and matches the working.
3. [B1] Factorise to obtain (cos θ – sin θ)(cos θ + 3 sin θ) = 0 OR Obtain 3 tan² θ – 2 tan θ - 1 = 0 OR Obtain (cos θ + sin θ)² = (2sin θ)²
4. [M1] Solve a quadratic in sin θ and cos θ / tan θ to obtain a value for θ. This leads to tan θ = 1 or tan θ = -1/3.
5. [A1] Obtain one correct value e.g. 45°
6. [A1] Obtain a second correct value e.g. 161.6° and no others in the interval. Mark answers in radians (0.785 and 2.82) as a misread. Accept awrt 161.6.