The identities and reduction rules that crack Paper 2 trig — simplifying, proving identities, using the CAST diagram, reducing angles and finding exact values — each worked in full, with a worksheet.
In Grade 11, trigonometry grows beyond right-angled triangles into identities (always true) and reduction formulae (rules for angles in different quadrants). This small toolkit lets you simplify expressions, prove identities and find exact values.
1The two identities you must know
The square identity is the star: rearrange it to sin2θ = 1 − cos2θ or cos2θ = 1 − sin2θ — that's how most 'prove the identity' questions are cracked.
2Type 1: Simplify an expression
- 1Cancel one sinθ: (sinθ·cosθ)/sin2θ = cosθ/sinθ.sin2θ = sinθ × sinθ.
- 2Recognise cosθ/sinθ = 1/tanθ.It's the reciprocal of tanθ = sinθ/cosθ.
3Type 2: Prove an identity
Work on the more complicated side only, using the identities, until it equals the other side.
- 1Left side: 1 − cos2θ = sin2θ.Rearranged square identity.
- 2So LHS = sin2θ / cosθ = sinθ · (sinθ/cosθ).Split the fraction.
- 3= sinθ · tanθ = RHS. ✓Since tanθ = sinθ/cosθ.
4Type 3: CAST: sign of a ratio by quadrant
CAST tells you which ratios are positive in each quadrant: 1st All positive, 2nd only Sine, 3rd only Tangent, 4th only Cosine.
- 1In the 4th quadrant only cosine is positive.CAST — the 'C' quadrant.
- 2So sin θ is negative.sin2θ = 1 − 0,36 = 0,64 → sinθ = −0,8 (negative in Q4).
5Type 4: Reduction formulae
Reduction formulae rewrite the ratio of a large or negative angle in terms of an acute angle. Key ones: sin(180° − θ) = sinθ, cos(180° − θ) = −cosθ, sin(360° − θ) = −sinθ, cos(−θ) = cosθ.
- 1180° − θ lands in the 2nd quadrant, where sine is positive.CAST: Q2 is 'S'.
- 2So sin(180° − θ) = sin θ.Same acute reference angle, positive sign.
6Type 5: Exact values using special angles
- 1120° = 180° − 60°, in the 2nd quadrant where cosine is negative.Reduce to the acute reference angle 60°.
- 2cos 120° = −cos 60° = −12.cos 60° = 12; the sign is negative in Q2.
Keep a single summary page of the identities, CAST and reduction rules and revise it weekly. In the exam, recognising which rule to use is 90% of the battle.
sin2θ means (sinθ)2, not sin(θ2). The square is on the whole ratio, not the angle.
This toolkit reappears throughout Grade 12 (identities, general solutions, 2D/3D problems), so the effort here pays off directly in matric. Practise every type below.
Practice exercises
Work each one out, then click to reveal the answer.
- 1Simplify sinθ · tanθ · cosθ. (simplify)
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sinθ·(sinθ/cosθ)·cosθ = sin2θ - 2Simplify (1 − sin2θ). (identity)
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cos2θ - 3Simplify cos2θ ⁄ (1 − sin2θ). (identity)
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cos2θ/cos2θ = 1 - 4Prove tanθ · cosθ = sinθ. (prove)
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(sinθ/cosθ)·cosθ = sinθ ✓ - 5In which quadrant are only sine positive? (CAST)
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2nd quadrant - 6sinθ = 0,5, θ in Q2. Is cosθ + or −? (CAST)
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Q2 → cosine negative - 7Simplify cos(180° − θ). (reduction)
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−cosθ - 8Simplify sin(360° − θ). (reduction)
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−sinθ - 9Exact value of sin 150°. (exact)
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150° = 180−30, Q2 sine + → sin30° = 12 - 10Exact value of cos 210°. (exact)
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210° = 180+30, Q3 cosine − → −cos30° = −3⁄2
Now practise it
Download Grade 11 past papers and worksheets on this topic.