Prove that 2 sin θ cosec 2θ ≡ sec θ.

Mark scheme

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

1. [M1] Express left-hand side in terms of sin θ and cos θ using cosec 2θ = 1/sin 2θ
2. [A1] Obtain 1/cos θ and confirm sec θ Answer given – necessary detail needed.
3. [*M1] Attempt to obtain quadratic equation in sec θ or cos θ only
4. [A1] Obtain sec²θ-1 + (7/2)sec θ = 8 involving one trigonometric ratio Or equivalent, may be unsimplified, but reduce to 2sec²θ+7secθ-18=0 or 18cos²θ-7cosθ-2=0.
5. [DM1] Attempt to solve 3-term quadratic equation for sec θ, using a correct method, to find at least one value of θ
6. [A1] Obtain any two of the four correct solutions ±0.952, ±1.76 Or greater accuracy.
7. [A1] Obtain remaining two correct solutions Or greater accuracy; and no others between –π and π.
8. [B1] Identify integrand as 2sec²(½x)
9. [M1] Integrate ksec²(½x) to obtain 2k tan(½x)
10. [A1] Obtain correct 4 tan(½x) Condone omission of ... + c.