Compound & Double Angle Identities (Grade 12)

The formulae for sin(A ± B) and cos(A ± B), the three forms of cos2A, using them to find exact values without a calculator, and proving identities.

1The compound angle formulae

cos(A − B) = cosA cosB + sinA sinB
cos(A + B) = cosA cosB − sinA sinB

sin(A + B) = sinA cosB + cosA sinB
sin(A − B) = sinA cosB − cosA sinB

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Remember the sign rule: cos changes the sign, sin keeps it. cos(A + B) has a minus in the middle; sin(A + B) has a plus.

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sin(A + B) is not sinA + sinB. Test it: sin(30° + 60°) = sin90° = 1, but sin30° + sin60° = 0,5 + 0,866 = 1,366.

2Exact values without a calculator

Worked ExampleDetermine the exact value of cos75°

1
Write 75° using two special angles: 75° = 45° + 30°.
Both are on the special-angle table.
2
cos75° = cos45°cos30° − sin45°sin30°.
Compound formula for cos(A + B).
3
= 12 × 32 − 12 × 12.
4
= 3 − 122.
Common denominator.
5
= 6 − 24.
Rationalise. Check on a calculator: 0,2588. ✓

3The double angle formulae

sin2A = 2 sinA cosA

cos2A = cos²A − sin²A
cos2A = 2cos²A − 1
cos2A = 1 − 2sin²A

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cos2A has three forms and choosing the right one is the whole skill. If the question also mentions sinA, use 1 − 2sin²A. If it mentions cosA, use 2cos²A − 1.

Worked ExampleIf sinθ = 35 and θ is acute, determine cos2θ

1
Use cos2θ = 1 − 2sin²θ.
We were given sinθ, so pick the form with only sin in it.
2
= 1 − 2(35)².
3
= 1 − 2(925) = 1 − 1825.
4
= 725.
No need to find cosθ at all.

4Proving an identity

Work down one side only until it equals the other. Never move terms across the equals sign — you are not solving an equation.

Worked ExampleProve that sin2A1 + cos2A = tanA

1
LHS = 2 sinA cosA1 + (2cos²A − 1).
Expand sin2A, and choose the cos2A form that will cancel the 1.
2
= 2 sinA cosA2cos²A.
The +1 and −1 cancel — that is why this form was chosen.
3
= sinAcosA.
Cancel 2cosA top and bottom.
4
= tanA = RHS. ✓

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Expand cos(A − B).
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cosA cosB + sinA sinB
2
Expand sin(A + B).
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sinA cosB + cosA sinB
3
Determine the exact value of cos75° without a calculator.
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cos(45°+30°) = 6 − 24
4
Determine the exact value of sin15° without a calculator.
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sin(45°−30°) = 6 − 24
5
Write down the three forms of cos2A.
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cos²A − sin²A, 2cos²A − 1, 1 − 2sin²A
6
If sinθ = 35 and θ is acute, determine cos2θ.
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1 − 2(925) = 725
7
If sinθ = 35 and θ is acute, determine sin2θ.
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cosθ = 45, so 2(35)(45) = 2425
8
Prove that sin2A1 + cos2A = tanA
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2sinAcosA2cos²A = sinAcosA = tanA
9
Simplify 2 sin15° cos15°.
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= sin30° = 12
10
Why is sin(A + B) ≠ sinA + sinB?
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Test A = 30°, B = 60°: sin90° = 1 but sin30° + sin60° ≈ 1,37, so they are not equal.