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0606/13 Oct/Nov 2025, Question 6: Trig Identities and Proofs

Cambridge IGCSE Additional Mathematics 0606 · Paper 13 (Non-calculator) · Non-calculator · 7 marks

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The question, in short

Use an equilateral triangle with side a and midpoint M to find sec 30°, then prove an identity involving sec x.

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

Part (a) · 3 marks

From the equilateral triangle diagram with side a and M the midpoint of AC, find sec 30°.

  1. 1AM = a/2
  2. 2BM² = a² − a²/4 = 3a²/4, so BM = (√3/2)a
  3. 3Angle ABM = 30°, so cos 30° = BM/AB = √3/2
  4. 4sec 30° = 1/cos 30° = 2/√3 = 2√3/3

Answersec 30° = 2/√3 or 2√3/3

Part (b) · 4 marks

Show that 1/(sec x − 1) + 1/(sec x + 1) simplifies to 2 cosec x cot x.

  1. 11/(sec x − 1) + 1/(sec x + 1) = 2 sec x/((sec x − 1)(sec x + 1))
  2. 2= 2 sec x/(sec²x − 1)
  3. 3= 2 sec x/tan²x (since sec²x = 1 + tan²x)
  4. 4= (2/cos x) × (cos²x/sin²x) = 2 cos x/sin²x
  5. 5= 2 × (1/sin x) × (cos x/sin x) = 2 cosec x cot x

Answer1/(sec x − 1) + 1/(sec x + 1) = 2 cosec x cot x shown

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