Sum and Difference Identities

Core Idea

The sum and difference identities express sin(A +/- B), cos(A +/- B), and tan(A +/- B) in terms of sines and cosines of A and B individually. For example, sin(A + B) = sin(A)cos(B) + cos(A)sin(B). These identities are the engine behind many others (double angle, half angle) and are used to find exact values of non-standard angles, simplify expressions, and solve equations.

How It's Best Learned

Derive the cosine difference formula geometrically using distance on the unit circle, then obtain the rest algebraically. Practice computing exact values like sin(75) = sin(45 + 30). Use the identities to derive double-angle formulas as a natural next step.

Common Misconceptions

• Believing sin(A + B) = sin(A) + sin(B): this is the most common and damaging error.
• Mixing up the sign patterns between sine and cosine sum/difference formulas.
• Forgetting the tangent sum formula or not recognizing when to use it.

Explainer

From your work with the unit circle and Pythagorean identities, you know that sin and cos describe coordinates on a unit circle, and that sin²θ + cos²θ = 1 no matter what angle θ is. The sum and difference identities extend this by answering a natural question: if I know sin A, cos A, sin B, and cos B separately, can I compute sin(A + B) without needing a calculator? The answer is yes — and the resulting formulas are among the most-used in all of trigonometry.

The four core identities are:

• sin(A + B) = sin A cos B + cos A sin B
• sin(A − B) = sin A cos B − cos A sin B
• cos(A + B) = cos A cos B − sin A sin B
• cos(A − B) = cos A cos B + sin A sin B

Notice the sign pattern: for cosine, the sign *flips* (cos of a sum has a minus sign); for sine, the sign *matches* (sin of a sum has a plus sign). A useful mnemonic is "cosine changes sign, sine stays the same as the ±." The most important thing to internalize is that sin(A + B) ≠ sin A + sin B. Sine is not a linear function — it does not distribute over addition. A quick counterexample: sin(30° + 60°) = sin 90° = 1, but sin 30° + sin 60° = 0.5 + 0.866 = 1.366 ≠ 1. If you try to distribute sine over a sum, you will consistently get wrong answers.

One of the best uses of these formulas is computing exact values of angles not on the standard unit circle. For example, sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4. You have turned an unfamiliar angle into a combination of the 30-60-90 and 45-45-90 triangles you already know cold. Similarly, cos 15° = cos(45° − 30°) = cos 45° cos 30° + sin 45° sin 30° = (√6 + √2)/4. Many exam problems give you unusual angles precisely because they expect you to decompose them this way.

These identities are also the engine for deriving everything that follows. Set B = A in the sine sum formula and you immediately get sin 2A = 2 sin A cos A — the double angle formula. Set B = A in the cosine sum formula to get cos 2A = cos²A − sin²A. The half-angle formulas follow by solving for cos²A and sin²A. Every subsequent identity in trigonometry either is a sum/difference identity in disguise or is derived from one. Treating these four formulas as your foundation, rather than memorizing each derived identity separately, is a far more efficient strategy — and understanding the derivation once makes reconstructing a forgotten formula trivial.