Show that 3 tan²θ + 5 sin²θ = (8 sin²θ - 5 sin⁴θ) / (1 - sin²θ)

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2025). The mark or mark type is shown in brackets.

1. [M1] Use of tan²θ = sin²θ / cos²θ
2. [M1] Relevant use of cos²θ = 1 - sin²θ at least once
3. [A1] 8sin²θ - 5sin⁴θ / 1 - sin²θ AG All necessary detail needed.
4. [*M1] Attempt to solve their 5sin⁴θ - 17sin²θ + 9 = 0 using a "correct" method Allow ± errors in arriving at their quadratic in sin²θ. This can be implied by either [sin²θ =] 2.744 or 0.6559.
5. [A1] sinθ = [±] 0.81[0] or √(17-√109)/10 Condone inclusion of sinθ=1.65 for this mark. Allow √(17±√109)/10. This mark can be implied by correct values.
6. [DM1] Their 54.1, and 180-their54.1 or 180+their 54.1 A correct method for obtaining a second angle within the range 0 < their 54.1 < 90.
7. [A1] 54.1, 125.9, 234.1 AWRT A0 for additional values between 0° and 270°.