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 \pm B), \cos(A \pm B), and \tan(A \pm B) in terms of \sin A, \cos A, \sin B, and \cos B.

What happens when you combine two rotations? If you rotate by angle A and then by angle B, 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.

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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: These formulas decompose trig functions of combined angles into products of trig functions of individual angles. They are the foundation for double-angle, half-angle, and product-to-sum formulas.

Common stuck point: The sign pattern in the cosine formula is opposite to what you might expect: \cos(A + B) has a MINUS sign between the terms, while \cos(A - B) has a PLUS sign.

Sense of Study hint: Verify by plugging in known angles: check that cos(60) = cos(30+30) gives the right answer using the formula.

Worked Examples

Example 1

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

Solution

  1. 1
    Write 75° = 45° + 30°.
  2. 2
    Apply the cosine sum formula: \cos(A+B) = \cos A \cos B - \sin A \sin B.
  3. 3
    Substitute: \cos(75°) = \cos(45°)\cos(30°) - \sin(45°)\sin(30°).
  4. 4
    = \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}.

Answer

\cos(75°) = \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).

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°) using a difference identity.

Example 2

hard
If \sin(\alpha) = \frac{4}{5} with \alpha in QI and \cos(\beta) = -\frac{5}{13} with \beta in QII, find \cos(\alpha + \beta).
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Background Knowledge

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

trig identities pythagoreantrigonometric functions