Sum and Difference Identities Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Sum and Difference Identities.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Formulas that express sin⁡(A±B)\sin(A \pm B), cos⁡(A±B)\cos(A \pm B), and tan⁡(A±B)\tan(A \pm B) in terms of sin⁡A\sin A, cos⁡A\cos A, sin⁡B\sin B, and cos⁡B\cos B.

What happens when you combine two rotations? If you rotate by angle AA and then by angle BB, the result involves both angles interacting. The sum and difference formulas tell you exactly how the trig values of two separate angles combine. They're like a multiplication rule for rotations—the result isn't simply adding the trig values, but mixing sines and cosines together.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: sin⁡(A±B)\sin(A\pm B) and cos⁡(A±B)\cos(A\pm B) expand into cross products of the separate angles' sines and cosines.

Common stuck point: The procedure for sum and difference identities is the easy part; the trap is splitting cos⁡(A+B)\cos(A+B) as cos⁡A+cos⁡B\cos A+\cos B. Asking "Is the argument of the trig function a sum or difference of two angles I want to break apart?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the argument of the trig function a sum or difference of two angles I want to break apart?

Worked Examples

Example 1

easy
Find the exact value of cos⁡(75°)\cos(75°) using the sum identity.

Answer

cos⁡(75°)=6−24\cos(75°) = \frac{\sqrt{6} - \sqrt{2}}{4}

First step

1
Write 75°=45°+30°75° = 45° + 30°.

Full solution

  1. 2
    Apply the cosine sum formula: cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A+B) = \cos A \cos B - \sin A \sin B.
  2. 3
    Substitute: cos⁡(75°)=cos⁡(45°)cos⁡(30°)−sin⁡(45°)sin⁡(30°)\cos(75°) = \cos(45°)\cos(30°) - \sin(45°)\sin(30°).
  3. 4
    =22⋅32−22⋅12=64−24=6−24= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}.
The sum and difference identities let us find exact values for angles that are sums or differences of standard angles (30°, 45°, 60°). Here we decomposed 75° as 45° + 30° and applied the cosine addition formula.

Example 2

medium
Simplify sin⁡(x+y)+sin⁡(x−y)\sin(x + y) + \sin(x - y).

Example 3

medium
Show that sin⁡(x+π/2)=cos⁡x\sin(x + \pi/2) = \cos x.

Example 4

hard
If sin⁡A=45\sin A = \tfrac{4}{5} in QI and cos⁡B=817\cos B = \tfrac{8}{17} in QIV, find sin⁡(A+B)\sin(A + B).

Example 5

hard
Prove the product-to-sum formula sin⁡Acos⁡B=12[sin⁡(A+B)+sin⁡(A−B)]\sin A\cos B = \tfrac{1}{2}[\sin(A+B) + \sin(A-B)].

Example 6

hard
Show that tan⁡(A+B)+tan⁡(A−B)=2sin⁡2Acos⁡2A+cos⁡2B\tan(A + B) + \tan(A - B) = \dfrac{2\sin 2A}{\cos 2A + \cos 2B}.

Example 7

challenge
Given sin⁡A=35\sin A = \tfrac{3}{5} in QII and cos⁡B=−1213\cos B = -\tfrac{12}{13} in QII, find tan⁡(A+B)\tan(A + B).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Find the exact value of tan⁡(15°)\tan(15°) using a difference identity.

Example 2

hard
If sin⁡(α)=45\sin(\alpha) = \frac{4}{5} with α\alpha in QI and cos⁡(β)=−513\cos(\beta) = -\frac{5}{13} with β\beta in QII, find cos⁡(α+β)\cos(\alpha + \beta).

Example 3

easy
State the formula for sin⁡(A+B)\sin(A + B).

Example 4

easy
State the formula for cos⁡(A+B)\cos(A + B).

Example 5

easy
State the formula for cos⁡(A−B)\cos(A - B).

Example 6

easy
State the formula for sin⁡(A−B)\sin(A - B).

Example 7

easy
Express sin⁡(90°−θ)\sin(90° - \theta) using the difference formula.

Example 8

easy
Express cos⁡(θ+90°)\cos(\theta + 90°) using the sum formula.

Example 9

easy
Is sin⁡(A+B)=sin⁡A+sin⁡B\sin(A + B) = \sin A + \sin B true in general? Answer yes or no.

Example 10

easy
Write tan⁡(A+B)\tan(A + B) in formula form.

Example 11

medium
Compute cos⁡75°\cos 75° exactly using 75°=45°+30°75° = 45° + 30°.

Example 12

medium
Compute sin⁡15°\sin 15° exactly using 15°=45°−30°15° = 45° - 30°.

Example 13

medium
Compute sin⁡75°\sin 75° exactly using 75°=45°+30°75° = 45° + 30°.

Example 14

medium
If sin⁡A=35\sin A = \frac35 (Q I) and cos⁡B=513\cos B = \frac{5}{13} (Q I), find sin⁡(A+B)\sin(A + B).

Example 15

medium
If sin⁡A=35\sin A = \frac35 (Q I) and cos⁡B=513\cos B = \frac{5}{13} (Q I), find cos⁡(A+B)\cos(A + B).

Example 16

medium
Simplify sin⁡50°cos⁡20°−cos⁡50°sin⁡20°\sin 50°\cos 20° - \cos 50°\sin 20°.

Example 17

medium
Compute tan⁡75°\tan 75° exactly using 75°=45°+30°75° = 45° + 30°.

Example 18

medium
Simplify cos⁡40°cos⁡10°+sin⁡40°sin⁡10°\cos 40°\cos 10° + \sin 40°\sin 10°.

Example 19

medium
Compute cos⁡15°\cos 15° exactly using 15°=45°−30°15° = 45° - 30°.

Example 20

challenge
Prove that sin⁡(A+B)+sin⁡(A−B)=2sin⁡Acos⁡B\sin(A + B) + \sin(A - B) = 2\sin A\cos B.

Example 21

challenge
Given tan⁡A=2\tan A = 2 and tan⁡B=3\tan B = 3, find tan⁡(A+B)\tan(A + B) and explain the geometric meaning.

Example 22

challenge
Show that cos⁡(A+B)cos⁡(A−B)=cos⁡2A−sin⁡2B\cos(A + B)\cos(A - B) = \cos^2 A - \sin^2 B.

Example 23

easy
Compute sin⁡30°cos⁡60°+cos⁡30°sin⁡60°\sin 30°\cos 60° + \cos 30°\sin 60°.

Example 24

easy
Compute cos⁡35°cos⁡25°−sin⁡35°sin⁡25°\cos 35°\cos 25° - \sin 35°\sin 25°.

Example 25

easy
Compute sin⁡70°cos⁡10°−cos⁡70°sin⁡10°\sin 70°\cos 10° - \cos 70°\sin 10°.

Example 26

easy
Express sin⁡(x+π)\sin(x + \pi) using the sum formula.

Example 27

easy
Express cos⁡(π−x)\cos(\pi - x) using the difference formula.

Example 28

medium
Compute tan⁡(45°−30°)\tan(45° - 30°) exactly.

Example 29

medium
Compute sin⁡105°\sin 105° exactly using 105°=60°+45°105° = 60° + 45°.

Example 30

medium
If sin⁡A=35\sin A = \tfrac{3}{5} in QI and sin⁡B=1213\sin B = \tfrac{12}{13} in QI, find sin⁡(A−B)\sin(A - B).

Example 31

medium
Simplify cos⁡(x+y)cos⁡y+sin⁡(x+y)sin⁡y\cos(x + y)\cos y + \sin(x + y)\sin y.

Example 32

medium
If cos⁡A=−35\cos A = -\tfrac{3}{5} in QII and sin⁡B=−513\sin B = -\tfrac{5}{13} in QIII, find cos⁡(A−B)\cos(A - B).

Example 33

medium
Simplify sin⁡(A+B)+sin⁡(A−B)cos⁡(A+B)+cos⁡(A−B)\dfrac{\sin(A+B) + \sin(A-B)}{\cos(A+B) + \cos(A-B)}.

Example 34

medium
Compute tan⁡105°\tan 105° exactly using 105°=60°+45°105° = 60° + 45°.

Example 35

medium
Express cos⁡(x−π/3)\cos(x - \pi/3) in the form acos⁡x+bsin⁡xa\cos x + b\sin x.

Example 36

hard
Simplify sin⁡(x+y)cos⁡y−cos⁡(x+y)sin⁡y\sin(x + y)\cos y - \cos(x + y)\sin y.

Example 37

hard
Simplify cos⁡(A−B)−cos⁡(A+B)sin⁡(A+B)+sin⁡(A−B)\dfrac{\cos(A-B) - \cos(A+B)}{\sin(A+B) + \sin(A-B)}.

Example 38

hard
If tan⁡A=13\tan A = \tfrac{1}{3} and tan⁡B=12\tan B = \tfrac{1}{2}, find A+BA + B (in radians, with A,BA, B in QI).

Example 39

hard
Express sin⁡x+cos⁡x\sin x + \cos x in the form Rsin⁡(x+ϕ)R\sin(x + \phi) with R>0R > 0, ϕ∈(−π/2,π/2)\phi \in (-\pi/2, \pi/2).

Example 40

challenge
In a triangle ABCABC, tan⁡A+tan⁡B+tan⁡C=tan⁡Atan⁡Btan⁡C\tan A + \tan B + \tan C = \tan A\tan B\tan C. Prove this using the sum formula.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

trig identities pythagoreantrigonometric functions