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0606/11 May/Jun 2025, Question 10: Trig Identities and Proofs

Cambridge IGCSE Additional Mathematics 0606 · Paper 11 (Non-calculator) · Non-calculator · 7 marks

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The question, in short

Simplify a trigonometric expression involving sin, cosec and tan to show it equals sec θ, then express sin x in terms of α given sec x = α.

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

Part (a) · 4 marks

For 0 ≤ θ < π/2, show that sin θ ÷ √(cosec²θ − 1) plus 1 ÷ √(1 + tan²θ) simplifies to sec θ.

  1. 1cosec²θ − 1 = cot²θ and 1 + tan²θ = sec²θ
  2. 2Expression = sin θ/√(cot²θ) + 1/√(sec²θ)
  3. 3= sin θ tan θ + cos θ (all positive for 0 ≤ θ < π/2)
  4. 4= sin θ × sin θ/cos θ + cos θ
  5. 5= (sin²θ + cos²θ)/cos θ = 1/cos θ = sec θ

Answersin θ/√(cosec²θ − 1) + 1/√(1 + tan²θ) = sec θ shown

Part (b) · 3 marks

Given sec x = α with 3π/2 < x ≤ 2π, find sin x in terms of α.

  1. 1cos x = 1/α
  2. 2sin²x + (1/α)² = 1
  3. 3sin x = ±√(1 − 1/α²)
  4. 4For 3π/2 < x ≤ 2π, sin x ≤ 0, so sin x = −√(1 − 1/α²)

Answersin x = −√(1 − 1/α²)

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Also searched as: 0606/11/M/J/25 · 0606_s25_qp_11 · 0606 s25 11 · question 10.
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