Show that cos⁴θ - sin⁴θ ≡ cos 2θ.

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

1. [*M1] Factorise LHS using difference of 2 squares. ((cos²θ - sin²θ)(cos²θ + sin²θ))
2. [DM1] Simplify. cos²θ + sin²θ = 1 must be seen or implied, e.g. (cos²θ-sin²θ)(cos²θ+sin²θ) = (cos²θ-sin²θ).
3. [A1] Obtain cos⁴θ - sin⁴θ = cos 2θ from correct working. AG.
4. [M1] Use part (a) and correct double angle formula to obtain expression involving sin² 2θ or [cos² 2θ]. [cos⁴θ - sin⁴θ + 4sin²θcos²θ] = [cos 2θ + sin² 2θ]. Allow BOD for 2 sin² 2θ if sin 2θ = 2sinθcosθ seen.
5. [B1] [cos 2θ] = (1/2) + (1/2)sin 2θ. Seen or implied.
6. [M1] Use of correct double angle formula on second part of the integral to obtain a form that can be integrated directly. e.g. ∫sin² 2θ dθ = ∫(1-cos 4θ)/2 dθ. Condone a mixture of x and θ.
7. [A1] Obtain (1/2)θ - (1/4)sin 4θ. Condone a mixture of x and θ.
8. [M1] Correct use of limits ± π/8 in an expression of the form pθ + q sin 2θ + r sin 4θ and evaluate the trig.
9. [A1] Obtain √2/4 + π/16 - 1/8. ISW. Or exact equivalent from exact working.