Tangent 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 tangent sum and difference works
Write tan(A + B) as sin(A + B) over cos(A + B), expand both with the sum rules, then divide every term on the top and bottom by cos A cos B. Each piece collapses into a tangent, the cosines vanish, and what is left is the compact fraction shown. The denominator picks up the opposite sign because the cosine expansion had a minus.
What each symbol means
$A,B$ are angles.
Tangent sum and difference: when it holds
The component tangents and final denominator must be defined and nonzero.
When it stops applying
When A + B reaches 90 degrees the product tan A tan B equals 1 and the denominator becomes 0. Try A = B = 45 degrees: the fraction reads 2/0. That is not a flaw, it is the formula reporting that tan 90 degrees does not exist. It also fails if A or B is itself 90 degrees, since that tangent is missing from the start.
Tangent sum and difference: a worked example
$\tan75^\circ=(1+1/\sqrt3)/(1-1/\sqrt3)=2+\sqrt3$.
The mistake to avoid
What people do: Students use the same sign in the numerator and the denominator, writing the bottom as 1 + tan A tan B for a sum of angles.
Why it goes wrong: The two signs always oppose each other. Keeping them the same destroys the one feature that makes the formula behave at large angles.
Do this instead: Set the sign on top to match the angle operation, then flip it underneath. For tan 75 = tan(45 + 30), the top is 1 + 0.577 and the bottom is 1 − 0.577, giving 3.732, which equals 2 + √3.
Tangent 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.
- Check the conditions before substituting. The component tangents and final denominator must be defined and nonzero.
- 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 tangent 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 tangent sum and difference
Why does the bottom start with a 1?
That 1 is what cos A cos B turns into after you divide the cosine expansion by itself. It is not a constant someone chose; it is the leftover of a term that cancelled completely.
Can I use it when one of the angles is 90 degrees?
No, and you do not need to. Since tan 90 degrees is undefined, the formula has nothing to substitute. Use the shift rule tan(θ + 90°) = −cot θ for that case.
What does it mean when the denominator lands on zero?
It means the combined angle is a quarter turn away from where tangent is defined, so the true tangent is undefined and the graph has a vertical asymptote there. Treat the zero as an answer, not an error.
Is there a matching rule for cotangent?
There is, and it looks inside out: the product term moves to the top and the single cotangents move to the bottom. Most people skip memorizing it and just take the reciprocal of the tangent result.