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A-LevelMathematicsTrigonometry9709/12 · May/June 2014 · Q57 Marks

Prove the identity 1/cos θ - cos θ/(1 + sin θ) = tan θ.

  1. (i)Prove the identity 1/cos θ - cos θ/(1 + sin θ) = tan θ.[4]
  2. (ii)Solve the equation 1/cos θ - cos θ/(1 + sin θ) + 2 = 0 for 0° ≤ θ ≤ 360°.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2014). The mark or mark type is shown in brackets.

  1. [M1] Correct addition of fractions
  2. [M1 M1] LHS = (1+s-c²)/(c(1+s)) = (s²+s)/(c(1+s)) Use of s²+c²=1. (1+s)cancelled.
  3. [A1] = tan θ → answer given.
  4. [M1] Uses part (i). Allow tan θ = –2
  5. [A1 A1] θ = 116.6° or 296.6° Co. √ for 180°+ and no other solutions in the range.

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About this question

Topic
Trigonometry, A-Level Mathematics (syllabus 9709). Structured question, 7 marks.
Source
A-Level Mathematics, May/June 2014 Paper 1 Variant 2, question 5.

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