Sign in

Principle deep dive

Compound angle: sin/cos/tan(A ± B)

The base trig identity that unlocks double angle, product-to-sum, and most identity manipulation. Tagged across Trig Identities + Trig Equations.

questions in the bank
42
tagged HARD
24%
chapter spread
2
worked examples below
4

When to reach for it

Two angles α, β appear and you need a trig function of α ± β.

Why this principle matters

Compound angle formulas — sin(A ± B), cos(A ± B), tan(A ± B) — are the substrate of every non-trivial trig identity. NDA tests them directly (Trigonometric Identities chapter) and indirectly (Trig Equations, Properties of Triangle, Inverse Trig).

The leverage isn't in memorising the formulas — it's in spotting WHEN to apply them. NDA hides compound angle behind expressions like sin²(π/4 + θ) − sin²(π/4 − θ), or (cos 17° − sin 17°)/(cos 17° + sin 17°), or m tan α = n tan β with α and β complementary. Each of these reduces to one application of a compound-angle formula.

Three formulas to memorise: sin(A + B) = sin A cos B + cos A sin B; cos(A + B) = cos A cos B − sin A sin B; tan(A + B) = (tan A + tan B)/(1 − tan A tan B). The minus versions follow by sign flip. From these, double angle (set B = A) and half angle (set A = θ/2) fall out for free.

4 worked examples from the bank

Each example demonstrates the principle on a real past-year question. Click to reveal the answer, then the solution.

Example 1Trigonometric IdentitiesEASY
If 3cos⁡θ=4sin⁡θ3\cos\theta = 4\sin\theta, then what is the value of tan⁡(45°+θ)\tan(45°+\theta)?

[Q30 · Apr · 2021]

Example 2Trigonometric IdentitiesEASY
What is cos⁡17°−sin⁡17°cos⁡17°+sin⁡17°\dfrac{\cos17°-\sin17°}{\cos17°+\sin17°} equal to?

[Q30 · Sep · 2024]

Example 3Trigonometric IdentitiesMODERATE
What is sin⁡2(π4+θ)−sin⁡2(π4−θ)\sin^2\left(\frac{\pi}{4}+\theta\right)-\sin^2\left(\frac{\pi}{4}-\theta\right) equal to?

[Q33 · Apr · 2022]

Example 4Trigonometric IdentitiesHARD
If tan⁡α=17, sin⁡β=110; 0<α,β<π2\tan\alpha=\frac{1}{7},\ \sin\beta=\frac{1}{\sqrt{10}};\ 0<\alpha,\beta<\frac{\pi}{2}, then what is the value of cos⁡(α+2β)\cos(\alpha+2\beta)?

[Q30 · Sep · 2023]

Variants to recognise

Same principle, different surfaces. Pattern-match these on test day.

  • sin(A + B) and sin(A − B)

    The base. Add to get the product-to-sum identity; subtract to isolate sin A cos B.

  • cos(A ± B)

    Negative inside flips the sin·sin sign — a common source of sign errors. Verify carefully.

  • tan(A ± B) and tan(45° + θ)

    tan(45° + θ) = (1 + tan θ)/(1 − tan θ). Recognising this saves time on NDA's favourite trick.

  • Double angle (B = A)

    sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1; tan 2A = 2 tan A / (1 − tan²A).

Drill every compound angle: sin/cos/tan(a ± b) question

42 questions from the bank — paginated, with cart and Word-export support.

Drill the 42 questions

Related principles

Often combined with this one — drill these next if you found the examples above tractable.