A-LevelMathematicsTrigonometry9709/12 · May/June 2017 · Q36 Marks
Prove the identity (1/cos θ - tan θ)² = (1 - sin θ) / (1 + sin θ)
- (i)Prove the identity (1/cos θ - tan θ)² = (1 - sin θ) / (1 + sin θ)[3]
- (ii)Hence solve the equation (1/cos θ - tan θ)² = 1/2, for 0° ≤ θ ≤ 360°.[3]
Mark scheme
Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.
- LHS = (1/c - s/c)^2
- [M1] Eliminates tan by replacing with sin/cos leading to a function of sin and/or cos only.
- (1-s)^2 / c^2
- [M1] Uses s^2 + c^2 = 1 leading to a function of sin only.
- (1-s)^2 / (1-s^2) = (1-s)(1-s) / (1-s)(1+s) = (1-s) / (1+s)
- [A1] AG. Must show use of factors for A1.
- Uses part (i) → (1 - s) / (1 + s) = 1/2
- [M1] Uses part (i) to obtain s = k
- → s = 1/3
- θ = 19.5° or 160.5°
- [A1A1 FT] FT from error in 19.5°. Allow 0.340° (0.3398°) & 2.80(2) or 0.108π° & 0.892πº for A1 only. Extra answers in the range lose the second A1 if gained for 160.5°.
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