Cosine sum and difference is one of 5 angle formulas formulas in the trigonometry section of this library, and it is used at high school · ap level.
Why cosine sum and difference works
Cosine measures how far along the horizontal axis a rotated point lands. When you add a second rotation, part of the old vertical height tips over into the new horizontal reading and works against it, and that opposing contribution is why the sin A sin B term is subtracted while the cos A cos B term is added.
What each symbol means
$A,B$ are angles in the same unit.
Cosine sum and difference: when it holds
Cosine uses the opposite sign in the middle.
When it stops applying
Nothing in the identity breaks, but reading it backwards does. Because cosine is even, cos(A − B) and cos(B − A) are equal, so a value like 0.259 for cos 75 degrees cannot tell you whether the original difference was 75 or −75 degrees.
Cosine sum and difference: a worked example
$\cos75^\circ=\frac{\sqrt6-\sqrt2}{4}$.
The mistake to avoid
What people do: Students copy the sine pattern and keep a plus in the middle, writing cos(A + B) = cos A cos B + sin A sin B.
Why it goes wrong: Check it at A = B = 60 degrees. The plus version gives 0.25 + 0.75 = 1, claiming that cos 120 degrees is 1, when the true value is −0.5.
Do this instead: Remember that cosine flips the sign: plus between the angles means minus between the products. The correct line reads 0.25 − 0.75 = −0.5, which matches cos 120 degrees.
Cosine sum and difference: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $A,B$ are angles in the same unit.
- Check the conditions before substituting. Cosine uses the opposite sign in the middle.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most trigonometry slips.
Where this formula fits
- Subject
- Trigonometry formulas — 18 entries in this library
- Topic
- Angle formulas
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where cosine sum and difference comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Trigonometric Identities — the lesson behind this formula: simplify and prove relationships between trig functions.
- Trigonometry Calculator — check your substitution and the value it produces.
- Study trigonometry — the subject guide that explains the ideas these formulas compress.
- Precalculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 18 trigonometry formulas — the full grouped reference, or the complete formula library.
Questions about cosine sum and difference
How do I keep the flipped sign straight during a test?
Use the word opposite: cosine takes the opposite sign of the one between the angles. A quick check at 60 + 60 degrees settles it in seconds, since only the minus version returns the −0.5 you already know.
Why does the difference version end up with a plus?
The rule flips whichever sign it sees. Feed a minus between the angles into the formula and the middle becomes a plus, which is how cos 15 degrees works out to (√6 + √2)/4, about 0.966.
Does the order of A and B matter?
For the sum version, no, since swapping them just swaps the two products. For the difference version, the angles matter but the answer does not change sign, because cosine gives the same output for an angle and its negative.
Where does this identity get used later?
It is the engine behind the double-angle and half-angle rules, and it turns products of waves into sums in physics and signal work. Setting B = A immediately yields cos 2θ = cos²θ − sin²θ.