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0606/21 Oct/Nov 2025, Question 4: Trig Identities and Proofs

Cambridge IGCSE Additional Mathematics 0606 · Paper 21 (Calculator) · Calculator · 6 marks

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The question, in short

Prove an identity linking two fractions with sin x and cos x to 2 sec x, then use that result to solve a related equation in θ over 0° to 720°.

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

Part (a) · 3 marks

Combine the two fractions and simplify to demonstrate the result equals 2 sec x.

  1. 1Common denominator: ((1 − sin x)² + cos²x)/(cos x(1 − sin x))
  2. 2Numerator = 1 − 2 sin x + sin²x + cos²x = 2 − 2 sin x (using sin²x + cos²x = 1)
  3. 3= 2(1 − sin x)/(cos x(1 − sin x))
  4. 4= 2/cos x = 2 sec x

Answer(1 − sin x)/cos x + cos x/(1 − sin x) = 2 sec x shown

Part (b) · 3 marks

Apply the proven identity to the equation equal to 3 and find all θ values from 0° to 720°.

  1. 1By part (a): 2 sec(θ/2) = 3
  2. 2cos(θ/2) = 2/3
  3. 30° ≤ θ/2 ≤ 360°: θ/2 = 48.19° or 311.81°
  4. 4θ = 96.4° or 623.6°

Answerθ = 96.4°, 623.6°

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Also searched as: 0606/21/O/N/25 · 0606_w25_qp_21 · 0606 w25 21 · question 4.
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