A-LevelMathematicsTrigonometry9709/33 · Oct/Nov 2024 · Q56 Marks
Show that cos⁴θ - sin⁴θ - 4 sin²θcos²θ ≡ cos²2θ + cos 2θ - 1.
- (a)Show that cos⁴θ - sin⁴θ - 4 sin²θcos²θ ≡ cos²2θ + cos 2θ - 1.[3]
- (b)Solve the equation cos⁴α - sin⁴α = 4 sin²α cos²α for 0° ≤ α ≤ 180°.[3]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.
- [B1] (a) Express cos⁴θ - sin⁴θ as (cos²θ+sin²θ)(cos²θ-sin²θ) or rewrite 4sin²θcos²θ as sin² 2θ.
- [B1] (a) Simplify to cos 2θ – sin² 2θ.
- [B1] (a) Use sin²2θ=1-cos² 2θ to obtain cos² 2θ + cos2θ-1. AG
- [M1] (b) State a quadratic equation in cos 2α and solve for α (e.g., cos² 2α + cos2α-1=0).
- [A1] (b) Obtain α = 25.9° or α = 154.1°. (May be more accurate. Allow 154 for 154.1).
- [A1] (b) Obtain α = 25.9° and α = 154.1° and no others in range.
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