O-LevelAdditional MathematicsTrigonometry4037/12 · May/June 2017 · Q68 Marks
Show that cosec θ / (cot θ + tan θ) = cos θ.
- (i)Show that cosec θ / (cot θ + tan θ) = cos θ.[4]
- (ii)It is given that ∫[0 to a] (cosec 2θ / (cot 2θ + tan 2θ)) dθ = √3 / 4, where 0 < a < π/4. Using your answer to part (i) find the value of a, giving your answer in terms of π.[4]
Mark scheme
Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.
- [M1] for cotθ = cosθ/sinθ, tanθ = sinθ/cosθ, cosecθ = 1/sinθ
- [M1] dealing with the fractions correctly
- [M1] use of identity
- [A1] correct simplification, with all correct
- [B1] ∫cos 2θ dθ = (1/2)sin 2θ
- [M1] use of [ksin 2θ]^a_0 = √3/4 to obtain ksin 2a = √3/4
- [DM1] attempt to solve equation of the form k sin 2a = √3/4 with -1 < √3/(4k) < 1, must have a correct order of operations dealing with the double angle
- [A1] a = π/6 or 0.167π or better
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