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A-LevelMathematicsTrigonometry9709/32 · Oct/Nov 2021 · Q87 Marks

By first expanding (cos² θ + sin²θ)², show that cos⁴ θ + sin⁴θ = 1 − ½ sin² 2θ.

  1. (a)By first expanding (cos² θ + sin²θ)², show that cos⁴ θ + sin⁴θ = 1 − ½ sin² 2θ.[3]
  2. (b)Hence solve the equation cos⁴θ + sin⁴θ = ¾, for 0° < θ < 180°.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

  1. [B1] Expand the square and equate to 1
  2. [M1] Use correct double angle formula
  3. [A1] Obtain cos⁴θ + sin⁴θ = 1 − ½ sin² 2θ
  4. [M1] Use the identity and carry out a method for finding a root
  5. [A1] Obtain answer 35.3°
  6. [A1 FT] Obtain a second answer, e.g. 54.7°
  7. [A1 FT] Obtain the remaining answers, e.g. 144.7° and 125.3° and no others in the given interval

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About this question

Topic
Trigonometry, A-Level Mathematics (syllabus 9709). Structured question, 7 marks.
Source
A-Level Mathematics, Oct/Nov 2021 Paper 3 Variant 2, question 8.

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