When is tangent undefined in the tangent addition formula?

Tangent of any angle is undefined when the denominator equals zero. In \tan(A + B), this happens when 1 - \tan A \tan B = 0, which occurs when the angle sum equals 90°, 270°, or any odd multiple of 90°. Similarly, \tan(A - B) is undefined when 1 + \tan A \tan B = 0.

Why do the signs in the denominator flip compared to the numerator?

This comes from the definitions of sine and cosine. When you derive \tan(A + B) from \frac{\sin(A+B)}{\cos(A+B)}, the sine part gets a + in the numerator, but the cosine part gets a - in the denominator, which is why the formula flips signs.

Can I use the tangent addition formula if one angle is in radians and the other is in degrees?

No; you must convert both angles to the same unit first. A tangent value only makes sense for an angle in a specific unit, so mixing units will give you a completely wrong answer.

What do I do if I don't know the exact value of \tan A or \tan B?

If you know the sine and cosine of an angle, use \tan = \frac{\sin}{\cos} to find the tangent value. If you only have a decimal approximation, you can use it, but the exact answer will usually be an approximation too — for homework or exams, prefer exact values like \frac{5}{12} or 2 + \sqrt{3} if possible.

Tangent Addition Formula

Find the tangent of angle sums and differences using a single formula that combines the tangent values of each angle.

\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}

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What each symbol means

What Tangent Addition Formula takes

Symbol	Meaning
$A$	The first angle, measured in radians or degrees; if confused with $\tan A$ (a single value), you'll apply the formula to a tangent value instead of an angle.
$B$	The second angle being added to or subtracted from A, in the same units as A; mixing units (one in radians, one in degrees) makes the result incorrect.

When to use it

Use the tangent addition formula when you need to find the tangent of an angle that is the sum or difference of two known angles.

Level

Usually taught in: Pre-Calculus

Worked examples

1. Find the tangent of 75° from angle sum

Problem

Calculate

\tan(75°)

using the fact that

75° = 45° + 30°

. Given:

\tan(45°) = 1

and

\tan(30°) = \frac{\sqrt{3}}{3}

$\tan(75°) = \tan(45° + 30°)$
Express the target angle as a sum of two angles with known tangent values.
$\tan(75°) = \frac{\tan(45°) + \tan(30°)}{1 - \tan(45°)\tan(30°)} = \frac{1 + \frac{\sqrt{3}}{3}}{1 - 1 \cdot \frac{\sqrt{3}}{3}}$
Apply the tangent addition formula, then substitute $\tan(45°) = 1$ and $\tan(30°) = \frac{\sqrt{3}}{3}$ .
$\frac{1 + \frac{\sqrt{3}}{3}}{1 - \frac{\sqrt{3}}{3}} = \frac{\frac{3 + \sqrt{3}}{3}}{\frac{3 - \sqrt{3}}{3}} = \frac{3 + \sqrt{3}}{3 - \sqrt{3}}$
Combine fractions in the numerator and denominator.
$\frac{3 + \sqrt{3}}{3 - \sqrt{3}} \cdot \frac{3 + \sqrt{3}}{3 + \sqrt{3}} = \frac{(3 + \sqrt{3})^2}{9 - 3} = \frac{9 + 6\sqrt{3} + 3}{6} = \frac{12 + 6\sqrt{3}}{6} = 2 + \sqrt{3}$
Rationalize by multiplying by the conjugate, then simplify.

Answer: $2 + \sqrt{3}$

This is the classic application: find an exact value for a non-standard angle by combining standard angles. The tangent addition formula is the direct route here.

2. Subtract tangent values with fractions

Problem

Given that

\tan A = \frac{3}{4}

and

\tan B = \frac{1}{2}

, find

\tan(A - B)

$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$
Use the tangent subtraction formula with opposite signs: numerator subtracts, denominator adds.
$\tan(A - B) = \frac{\frac{3}{4} - \frac{1}{2}}{1 + \frac{3}{4} \cdot \frac{1}{2}} = \frac{\frac{3}{4} - \frac{2}{4}}{1 + \frac{3}{8}}$
Substitute the values and convert $\frac{1}{2}$ to $\frac{2}{4}$ to subtract from $\frac{3}{4}$ .
$\tan(A - B) = \frac{\frac{1}{4}}{\frac{8 + 3}{8}} = \frac{\frac{1}{4}}{\frac{11}{8}}$
Simplify the numerator to $\frac{1}{4}$ and the denominator to $\frac{11}{8}$ .
$\tan(A - B) = \frac{1}{4} \cdot \frac{8}{11} = \frac{8}{44} = \frac{2}{11}$
Divide fractions by multiplying by the reciprocal, then reduce.

Answer: $\frac{2}{11}$

This example uses fractions and the subtraction form, highlighting that the denominator signs flip: when subtracting angles, the denominator requires addition.

3. Construction job: combining slope angles

Problem

Two ramps are being joined end-to-end on a construction site. Ramp A has an inclination angle with tangent

\frac{5}{12}

, and Ramp B has tangent

\frac{2}{3}

. What is the tangent of the combined angle?

$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$
Use the addition formula since the ramps are placed end-to-end, creating a combined angle.
$\tan(A + B) = \frac{\frac{5}{12} + \frac{2}{3}}{1 - \frac{5}{12} \cdot \frac{2}{3}}$
Substitute the known tangent values from each ramp.
$\text{Numerator: } \frac{5}{12} + \frac{2}{3} = \frac{5}{12} + \frac{8}{12} = \frac{13}{12}$
Find a common denominator and add the numerator terms.
$\text{Denominator: } 1 - \frac{5}{12} \cdot \frac{2}{3} = 1 - \frac{10}{36} = 1 - \frac{5}{18} = \frac{18 - 5}{18} = \frac{13}{18}$
Multiply the tangent values, then subtract from 1.
$\tan(A + B) = \frac{\frac{13}{12}}{\frac{13}{18}} = \frac{13}{12} \cdot \frac{18}{13} = \frac{18}{12} = \frac{3}{2}$
Divide fractions and simplify.

Answer: $\frac{3}{2}$

This real-world application shows how the formula handles practical angle combinations. The tangent of the combined slope is $\frac{3}{2}$ , which the foreman can use to plan the final section.

Common mistakes

Where Tangent Addition Formula usually goes wrong

The mistake	Why it is wrong	The fix
Writing $\tan(A + B) = \tan A + \tan B$	This ignores the denominator of the formula and forgets that tangent is not additive.	Always use the full formula: $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$
Using the wrong sign in the denominator, like $\tan(A - B) = \frac{\tan A - \tan B}{1 - \tan A \tan B}$ instead of $\frac{\tan A - \tan B}{1 + \tan A \tan B}$	The signs in the denominator are opposite to those in the numerator; subtract in the top means add in the bottom, and vice versa.	Remember: numerator uses $\pm$ , denominator uses $\mp$ (opposite signs).
Plugging in $\tan(45°) = 45$ or using the degree measure directly in the formula.	The formula needs the actual tangent VALUE of each angle, not the angle itself.	First find or calculate $\tan A$ and $\tan B$ as numbers, then substitute those values.

Tips and when to use something else

Remember that the denominator has opposite signs from the numerator: if you add on top ( $+$ ), subtract in the bottom ( $-$ ); if you subtract on top ( $-$ ), add in the bottom ( $+$ ).
When the denominator equals zero — that is, when $1 - \tan A \tan B = 0$ or $1 + \tan A \tan B = 0$ — the angle sum or difference results in $90°$ or $270°$ (undefined tangent); stop and report that the tangent does not exist.
If both angles are given in degrees, keep them in degrees throughout; if both are in radians, work in radians. Never mix units.
For angles close to $45°$ or complicated expressions, consider using the Sine Addition Formula and Cosine Addition Formula instead, then computing $\tan = \frac{\sin}{\cos}$ — this is often cleaner than the direct formula.

Related formulas

Sine Addition FormulaThe Sine Addition Formula expands sine of a sum or difference into a product of sines and cosines, making it useful for exact angle calculations.Cosine Addition FormulaFind the cosine of an angle sum or difference by using the cosine and sine values of the individual angles—essential for solving trigonometric equations.Double Angle FormulasDouble angle formulas express trig functions of 2θ in terms of single angles, letting you simplify complex expressions and solve trigonometric equations.Solving Trigonometric EquationsFind all angles that produce a specific sine value by using inverse sine and the periodic nature of trigonometric functions.

Frequently asked questions

When is tangent undefined in the tangent addition formula?: Tangent of any angle is undefined when the denominator equals zero. In $\tan(A + B)$ , this happens when $1 - \tan A \tan B = 0$ , which occurs when the angle sum equals $90°$ , $270°$ , or any odd multiple of $90°$ . Similarly, $\tan(A - B)$ is undefined when $1 + \tan A \tan B = 0$ .
Why do the signs in the denominator flip compared to the numerator?: This comes from the definitions of sine and cosine. When you derive $\tan(A + B)$ from $\frac{\sin(A+B)}{\cos(A+B)}$ , the sine part gets a $+$ in the numerator, but the cosine part gets a $-$ in the denominator, which is why the formula flips signs.
Can I use the tangent addition formula if one angle is in radians and the other is in degrees?: No; you must convert both angles to the same unit first. A tangent value only makes sense for an angle in a specific unit, so mixing units will give you a completely wrong answer.
What do I do if I don't know the exact value of $\tan A$ or $\tan B$ ?: If you know the sine and cosine of an angle, use $\tan = \frac{\sin}{\cos}$ to find the tangent value. If you only have a decimal approximation, you can use it, but the exact answer will usually be an approximation too — for homework or exams, prefer exact values like $\frac{5}{12}$ or $2 + \sqrt{3}$ if possible.

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Reviewed 2026-09-18