Hence or otherwise solve the equation sec⁴2α-tan⁴ 2α = 2 tan² 2α sec² 2α for 0° < α < 180°.

Mark scheme

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

1. [M1] Form an equation in tan 2α. 1+2tan² 2α = 2tan² 2α(1+tan² 2α) Or multiply through by cos⁴ 2α to form an equation in sin 2α or cos 2α cos⁴ 2α + 2sin² 2α cos² 2α = 2sin⁴ 2α ⇒ sin⁴ 2α +2sin² 2α-1=0 or cos⁴ 2α-4cos² 2α +2 = 0
2. [M1] Solve for tan 2α or equivalent tan 2α = ±(1/√2)
3. [A1] Obtain one correct solution for α, e.g. 20.0(30..)°
4. [A1] Obtain a second correct value for α, e.g. 70.0° (69.9698..°)
5. [A1] Obtain solutions 110° (110.0) and 160° (160.0) for α, and no others in range