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4PM1/02 Jan 2020, Question 11: The Addition Formulae

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 13 marks

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The question, in short

Use trigonometric identities to rewrite an equation as tan A = k tan B, prove an identity in θ, and derive exact values for cos 15° and tan 255°.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 4 marks

Rewrite 3 sin(A − B) = sin(A + B) as tan A = k tan B and state the integer k.

  1. 13 sin A cos B − 3 cos A sin B = sin A cos B + cos A sin B
  2. 22 sin A cos B = 4 cos A sin B
  3. 3Divide by cos A cos B: 2 tan A = 4 tan B
  4. 4tan A = 2 tan B, so k = 2

Answertan A = 2 tan B, k = 2

Part (b) · 3 marks

For θ ≠ (2n + 1)π/2, prove that (cos⁴θ − sin⁴θ)/cos²θ equals 1 − tan²θ.

  1. 1cos⁴θ − sin⁴θ = (cos²θ − sin²θ)(cos²θ + sin²θ)
  2. 2cos²θ + sin²θ = 1
  3. 3So the expression = (cos²θ − sin²θ)/cos²θ
  4. 4= 1 − sin²θ/cos²θ = 1 − tan²θ

Answer(cos⁴θ − sin⁴θ)/cos²θ = 1 − tan²θ shown

Part (ci) · 2 marks

Using exact values at 30°, 45°, 60°, show that cos 15° = (√6 + √2)/4.

  1. 1cos 15° = cos(45° − 30°) = cos 45° cos 30° + sin 45° sin 30°
  2. 2= (√2/2)(√3/2) + (√2/2)(1/2)
  3. 3= √6/4 + √2/4 = (√6 + √2)/4

Answercos 15° = (√6 + √2)/4 shown

Part (cii) · 4 marks

Using exact values at 30°, 45°, 60°, show that tan 255° = (3 + √3)/(3 − √3).

  1. 1tan 255° = tan(255° − 180°) = tan 75°
  2. 2tan 75° = tan(30° + 45°) = (tan 30° + tan 45°)/(1 − tan 30° tan 45°)
  3. 3= (1/√3 + 1)/(1 − 1/√3)
  4. 4Multiply top and bottom by 3: (√3 + 3)/(3 − √3)
  5. 5= (3 + √3)/(3 − √3)

Answertan 255° = (3 + √3)/(3 − √3) shown

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