Sum and Difference Identities Calculator

The sign trap

sin(A + B) = sin A cos B + cos A sin B. Plus stays plus.

cos(A + B) = cos A cos B − sin A sin B. Plus becomes minus.

There is a reason rather than a rule to memorise. Cosine is largest at 0° and falls away in both directions, so adding two angles moves you further from the peak and the value drops faster than the individual cosines suggest. The minus is that drop.

Sanity check that always works: put B = 0. Then cos(A + 0) should be cos A, and the formula gives cos A × 1 − sin A × 0 = cos A. Correct. With a plus it would give cos A + 0, which also works — so use B = A instead: cos 2A should be cos²A − sin²A, and only the minus form gives that.

Where they come from

These are not arbitrary. They are what rotation does, and there are several routes to them.

Geometrically: draw two angles stacked, drop perpendiculars, and the identities fall out of similar triangles. This is the classical proof and it is genuinely visual.

By rotation matrices: rotating by A and then by B is the same as rotating by A + B. Multiplying the two matrices out and comparing entries gives both identities at once.

By Euler’s formula: e^(i(A+B)) = e^(iA) × e^(iB). Expanding both sides and matching real and imaginary parts gives cosine and sine respectively. This is the quickest derivation and the one that explains why the signs behave as they do — the minus comes from i².

Building exact values

The sixteen unit-circle angles are not the only ones with exact values. Any sum or difference of two of them has one too.

75° = 45° + 30°. So sin 75° = sin 45° cos 30° + cos 45° sin 30° = (√6 + √2)/4.

15° = 45° − 30°. So sin 15° = (√6 − √2)/4.

That extends the exact table to every multiple of 15°, and combining with the half-angle formulas extends it to 7.5° and beyond. The page gives the decimals; the surd form comes from doing the substitution by hand, which is what an exam is usually asking for.

The cofunction identities

Put A = 90° into the difference formulas and something useful falls out.

cos(90° − B) = cos 90° cos B + sin 90° sin B = 0 + sin B = sin B.

So cos(90° − θ) = sin θ, and by the same route sin(90° − θ) = cos θ. Each function of an angle equals the co-function of its complement.

This is where the “co” in cosine comes from: it is the complementary sine. The same holds for tangent and cotangent, and for secant and cosecant.

In a right triangle it is obvious: the two acute angles are complementary, and one angle’s opposite side is the other’s adjacent.

What they are for

Combining waves. Adding two sinusoids of the same frequency gives another sinusoid, and the sum formulas are what prove it and what give the resulting amplitude and phase. Every interference calculation rests on this.

Rotation. Rotating a point by A and then by B is one rotation by A + B, and the identities are that statement in coordinates. Graphics and robotics use them constantly.

Simplifying before calculus. An expression rewritten as a single trigonometric function of a compound angle is usually far easier to differentiate or integrate than the product it started as.

Deriving everything else. The double, half, product-to-sum and sum-to-product identities are all consequences of these five. Learning these makes the rest derivable rather than memorable.

Product to sum

Add the two cosine identities together and the sine terms cancel: cos(A − B) + cos(A + B) = 2 cos A cos B.

Rearranged, that is a product turned into a sum: cos A cos B = ½[cos(A − B) + cos(A + B)]. The other three product-to-sum identities come out the same way, by adding or subtracting the appropriate pair.

It matters because sums integrate and products do not. Turning cos 3x cos 5x into ½[cos 2x + cos 8x] converts an intractable integral into two immediate ones.

It is also the mathematics behind beats: multiplying two nearby frequencies produces a sum and a difference, and the difference is the slow throb you hear when two instruments are almost in tune.

Sources and methodology

The identities are foundational and standard; these are the references.

• National Council of Teachers of Mathematics — where the compound-angle identities are introduced
• American Mathematical Society — where the identities and their proofs are catalogued
• Double and Half Angle Calculator — these identities with the two angles set equal
• How we calculate — methodology

Method. Each identity is evaluated through its formula from sin A, cos A, sin B and cos B, and separately by computing the combined angle and taking its sine or cosine directly. Those are genuinely different routes through the arithmetic, so agreement between them is evidence rather than a restatement — and the page reports the largest disagreement so the reader can see it is rounding. The suite runs the comparison across four hundred and forty-one angle pairs. That engine is verified on every change against 71 hand-written assertions, including that all four sum and difference identities match a direct computation of the combined angle across four hundred and forty-one angle pairs. The count and the per-case breakdown are published on the formula verification page.

Related calculators

Where this goes next:

More in Math, or browse all calculators.

Read the guide

Setting B = A in these gives the double-angle formulas, and rearranging those gives the half-angle ones — the Double and Half Angle Calculator carries that step through with the ± resolved.