Trigonometric Identities & Equations

Sum & Difference Identities

Using sum and difference formulas to evaluate and simplify trig expressions.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Pythagorean Identities

A Tempting but False Shortcut

Before reading on, test a guess: is sin(A + B) the same as sin(A) + sin(B)? Try A = 30° and B = 60°. sin(30° + 60°) = sin(90°) = 1. Now compute sin(30°) + sin(60°) separately and compare.

sin(30°) + sin(60°) = 1/2 + √3/2 ≈ 1.366 — nowhere close to 1. Sine (and cosine) genuinely don't distribute over addition the way multiplication does; the real relationship is more involved.

Definition — Sum and Difference Identities

sin(A ± B) = sin A cos B ± cos A sin B. cos(A ± B) = cos A cos B ∓ sin A sin B (note the sign flips between the sine and cosine versions).

These formulas come from comparing two ways of measuring the same distance on the unit circle: the straight-line distance between the points at angles A and B can be computed directly using the angle difference (A − B) alone, or computed using each point's own coordinates, (cos A, sin A) and (cos B, sin B), through the distance formula. Setting those two distance calculations equal to each other and simplifying produces exactly the difference identity for cosine — from which the others can all be derived using symmetry and the Pythagorean identity.

Worked Example — Finding an Exact Value Using the Sum Identity

Find the exact value of cos(75°) using cos(45° + 30°). Apply the identity: cos(45° + 30°) = cos 45° cos 30° − sin 45° sin 30° = (√2/2)(√3/2) − (√2/2)(1/2) = √6/4 − √2/4 = (√6 − √2)/4.

Worked Example — Verifying the Result Numerically

Check (√6 − √2)/4 against a decimal approximation of cos(75°). √6 ≈ 2.449, √2 ≈ 1.414, so (√6 − √2)/4 ≈ 1.035/4 ≈ 0.259. Since cos(75°) ≈ 0.259 as well, the exact value checks out.

\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \qquad \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B

Tip

Memorize the sign pattern by noticing sine's formula keeps a matching sign (+ with +, − with −) while cosine's formula flips it — a quick way to check a formula written from memory.

Common Mistakes

Assuming sin(A + B) = sin A + sin B or cos(A + B) = cos A + cos B, distributing across the addition the way multiplication does.
Trig functions of a sum don't distribute — always use the actual sum identity, sin A cos B + cos A sin B, rather than adding the trig values directly.
Mixing up the sign pattern between the sine and cosine versions of the identity.
Sine's formula matches the sign in A ± B directly; cosine's formula flips it — double-check which formula is being used before applying the sign.

Key Takeaways

sin(A + B) and cos(A + B) are not the same as adding the individual sine or cosine values — they follow specific sum formulas instead.
These formulas come from comparing two ways of measuring the same distance between two points on the unit circle.
The sum and difference identities allow exact trig values to be found for angles built from familiar reference angles.

Summary

Sum and difference identities let exact trig values be computed for many more angles than the basic reference angles alone. The final lesson in this unit uses these identities to solve trigonometric equations.

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