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0606/22 Feb/Mar 2026, Question 9: Solving Trig Equations

Cambridge IGCSE Additional Mathematics 0606 · Paper 22 (Calculator) · Calculator · 12 marks

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The question, in short

Prove a trigonometric identity involving tan, cot and sec, then use it to solve two equations, one in degrees and one in radians.

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

Part (a) · 3 marks

Show that tan θ + cot θ + 2 sec θ simplifies to (1 + 2 sin θ) ÷ (sin θ cos θ).

  1. 1tan θ + cot θ + 2 sec θ = sin θ/cos θ + cos θ/sin θ + 2/cos θ
  2. 2= (sin²θ + cos²θ + 2 sin θ)/(sin θ cos θ)
  3. 3= (1 + 2 sin θ)/(sin θ cos θ), using sin²θ + cos²θ = 1

Answertan θ + cot θ + 2 sec θ = (1 + 2 sin θ)/(sin θ cos θ) shown

Part (b) · 4 marks

Using part a, find all x in −180° ≤ x ≤ 180° satisfying tan 2x + cot 2x + 2 sec 2x = 0.

  1. 1By (a): (1 + 2 sin 2x)/(sin 2x cos 2x) = 0, so sin 2x = −1/2
  2. 2−360° ≤ 2x ≤ 360°
  3. 32x = −150°, −30°, 210°, 330°
  4. 4x = −75°, −15°, 105°, 165°

Answerx = −75°, −15°, 105°, 165°

Part (c) · 5 marks

Find all x in radians, −1 < x < 3, satisfying 3 sec(x − 1) − 1 = tan²(x − 1).

  1. 1tan²(x − 1) = sec²(x − 1) − 1
  2. 23 sec(x − 1) − 1 = sec²(x − 1) − 1
  3. 3sec²(x − 1) − 3 sec(x − 1) = 0, so sec(x − 1) = 3 (sec ≠ 0)
  4. 4cos(x − 1) = 1/3, with −2 < x − 1 < 2
  5. 5x − 1 = 1.23 or −1.23
  6. 6x = 2.23 or x = −0.231

Answerx = −0.231, x = 2.23

Stuck on a step? Read the free lesson: Solving Trig Equations

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Also searched as: 0606/22/F/M/26 · 0606_m26_qp_22 · 0606 m26 22 · question 9.
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