A-LevelMathematicsIntegration9709/31 · May/June 2021 · Q46 Marks
Prove that (1 - cos 2θ) / (1 + cos 2θ) = tan² θ.
- (a)Prove that (1 - cos 2θ) / (1 + cos 2θ) = tan² θ.[2]
- (b)Hence find the exact value of ∫(from π/6 to π/3) ((1 - cos 2θ) / (1 + cos 2θ)) dθ.[4]
Mark scheme
Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.
- [M1] Use correct double angle formula or t-substitution twice
- [A1 AG] Obtain (1-cos 2θ) / (1+cos 2θ) = tan² θ from correct working
- [M1] Express tan² θ in terms of sec²θ
- [A1] Integrate and obtain terms tan θ − θ
- [M1] Substitute limits correctly in an integral of the form a tan θ + bθ, where ab ≠ 0
- [A1] Obtain answer (2√3 / 3) - (π / 6) or equivalent exact 2-term expression
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