Show that the equation cos 2x(tan² 2x + 3) + 3 = 0 can be expressed as 2 cos² 2x + 3 cos 2x + 1 = 0.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

1. EITHER: Uses tan²2x = sin²2x/cos²2x (M1) Replaces tan²2x by sin²2x/cos²2x not sin²2x/cos²x
2. Uses sin²2x = (1 - cos²2x) (M1) Replaces sin²2x by (1 – cos²2x)
3. → 2cos²2x + 3cos2x + 1 = 0 (A1) AG. All correct
4. OR: tan²2x = sec²2x - 1 (M1) Replaces tan²2x by sec²2x-1
5. sec²2x = 1/cos²2x (M1) Replaces sec²2x by 1/cos²2x
6. Multiply through by cos²2x and rearrange (Implied by M1 above)
7. → 2cos²2x + 3cos2x + 1 = 0 (A1) AG. All correct
8. cos 2x = -½, -1 (M1) Uses (i) to get values for cos 2x. Allow incorrect sign(s).
9. 2x = 120°, 240° or 2x = 180°1 (A1A1 FT) A1 for 60° or 120° FT for 180-1st answer
10. x = 60° or 120°
11. or x = 90° (A1) Any extra answer(s) in given range only penalise fourth mark so max 3/4.