CAIE A-Level Mathematics 3.3.2 Trigonometric Identities

CAIE A-Level Mathematics 3.3.2 Trigonometric Identities
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise proving and applying reciprocal, compound-angle, double-angle and R-form identities to simplify expressions and solve equations across restricted intervals.

How this is tested

  • rewrite sec, cosec and cot or expand compound and double angles in sine and cosine
  • transform a sin x + b cos x into an R form using R² = a² + b² and the correct α
  • use the derived identity to solve for every angle in range and reject extraneous branches

Question 7

Question 7(a)

(a)

Prove that cos⁡(θ+30∘)cos⁡(θ+60∘)≡143−12sin⁡2θ\cos \left(\theta+30^{\circ}\right) \cos \left(\theta+60^{\circ}\right) \equiv \frac{1}{4} \sqrt{3}-\frac{1}{2} \sin 2 \theta.

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Question 7(b)

(b)

Solve the equation 5cos⁡(2α+30∘)cos⁡(2α+60∘)=15 \cos \left(2 \alpha+30^{\circ}\right) \cos \left(2 \alpha+60^{\circ}\right)=1 for 0∘<α<90∘0^{\circ}<\alpha<90^{\circ}.

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Question 7(c)

(c)

Show that the exact value of cos⁡20∘cos⁡50∘+cos⁡40∘cos⁡70∘\cos 20^{\circ} \cos 50^{\circ}+\cos 40^{\circ} \cos 70^{\circ} is 123\frac{1}{2} \sqrt{3}.

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