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0606/23 May/Jun 2026, Question 6: Solving Trig Equations

Cambridge IGCSE Additional Mathematics 0606 · Paper 23 (Calculator) · Calculator · 8 marks

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The question, in short

Prove a trigonometric identity involving sin θ divided by sec θ minus tan θ plus sin θ divided by sec θ plus tan θ, then use the result to solve a related equation for θ between 0° and 360°.

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

Part (a) · 3 marks

Show that the sum of the two fractions simplifies to 2 tan θ.

  1. 1Common denominator: sin θ[(sec θ + tan θ) + (sec θ − tan θ)]/((sec θ − tan θ)(sec θ + tan θ))
  2. 2= 2 sin θ sec θ/(sec²θ − tan²θ)
  3. 3sec²θ − tan²θ = 1
  4. 4= 2 sin θ/cos θ = 2 tan θ

Answersin θ × 2 sec θ/(sec²θ − tan²θ) = 2 sin θ/cos θ = 2 tan θ — shown

Part (b) · 5 marks

Using part a, find all values of θ in the range 0° to 360° satisfying the given equation equal to cos θ.

  1. 12 tan θ = cos θ ⇒ 2 sin θ = cos²θ
  2. 22 sin θ = 1 − sin²θ ⇒ sin²θ + 2 sin θ − 1 = 0
  3. 3sin θ = −1 ± √2; −1 − √2 < −1 is impossible
  4. 4sin θ = √2 − 1 = 0.4142
  5. 5θ = 24.5° or 180° − 24.5° = 155.5°

Answerθ = 24.5°, 155.5°

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