What Is the Tangent Identity? (Trig Proofs)
The tangent sum identity is (\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}). It follows from the sine and cosine angle-sum formulas, not from adding two tangent values. The formula applies only when the tangent terms exist and its denominator is not zero. The same reasoning gives a related identity for a difference of angles.
If you are learning trigonometry, the formula can look like a rule to memorize without a reason. The useful part is the proof: it shows where the formula comes from and when you can safely use it. I often find that checking the conditions is as important as doing the algebra, because a formula cannot give a defined answer when its denominator is zero.
Tangent Identity Overview: Definition, Sum Formula, and Conditions
The tangent identity connects the tangent of a combined angle to the tangents of two angles. To use it correctly, start with the definition of tangent as sine divided by cosine. Then check that the relevant cosines and the formula’s denominator are nonzero.
For angles (A) and (B), the sum identity is
[ \boxed{\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}} ]
The subtraction identity is
[ \boxed{\tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}} ]
These are called tangent sum and difference identities, or tangent addition and subtraction formulas.
Tangent is the ratio of the sine of an angle to its cosine:
[ \tan x=\frac{\sin x}{\cos x} ]
This ratio is defined only when (\cos x\ne0). In degree measure, cosine is zero at (90^\circ+180^\circ k), where (k) is any integer. In radian measure, those angles are (\frac{\pi}{2}+k\pi).
For the sum identity, you need (\cos A\ne0), (\cos B\ne0), and (\cos(A+B)\ne0). The first two conditions make (\tan A) and (\tan B) defined. The last condition makes (\tan(A+B)) defined. When the first two conditions hold, the last one is equivalent to (1-\tan A\tan B\ne0).
| What to check | Sum identity requirement | Why it matters |
|---|---|---|
| (\tan A) exists | (\cos A\ne0) | Tangent divides by cosine |
| (\tan B) exists | (\cos B\ne0) | Tangent divides by cosine |
| (\tan(A+B)) exists | (\cos(A+B)\ne0) | The left side must be defined |
| Quotient denominator | (1-\tan A\tan B\ne0) | The right side must be defined |
The subtraction identity has the matching conditions: (\cos A\ne0), (\cos B\ne0), and (\cos(A-B)\ne0). Its denominator (1+\tan A\tan B) must not be zero.
Deriving the Identity: Angle-Sum Formulas and Algebra
The proof begins with two standard angle-sum formulas: one for sine and one for cosine. Write tangent as a sine-over-cosine ratio, substitute those formulas, and divide the numerator and denominator by (\cos A\cos B). This produces the tangent addition formula.
The required sine and cosine formulas are
[ \sin(A+B)=\sin A\cos B+\cos A\sin B ]
and
[ \cos(A+B)=\cos A\cos B-\sin A\sin B. ]
Now begin with the left side:
[ \tan(A+B)=\frac{\sin(A+B)}{\cos(A+B)} ]
Substitute the two angle-sum formulas:
[ \tan(A+B)= \frac{\sin A\cos B+\cos A\sin B} {\cos A\cos B-\sin A\sin B} ]
If (\cos A\ne0) and (\cos B\ne0), divide every term in the numerator and denominator by (\cos A\cos B):
[ \tan(A+B)= \frac{\frac{\sin A}{\cos A}+\frac{\sin B}{\cos B}} {1-\frac{\sin A}{\cos A}\frac{\sin B}{\cos B}} ]
Since (\frac{\sin A}{\cos A}=\tan A) and (\frac{\sin B}{\cos B}=\tan B), this becomes
[ \tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}. ]
That proves the formula wherever the expressions are defined. The subtraction proof follows the same steps, using (\sin(A-B)=\sin A\cos B-\cos A\sin B) and (\cos(A-B)=\cos A\cos B+\sin A\sin B).
Proof Skills: Domain Checks, Common Errors, and Practice
A proof is valid only when each division is allowed. In this derivation, dividing by (\cos A\cos B) requires both cosines to be nonzero. The final quotient also requires a nonzero denominator. Checking these facts prevents a correct-looking calculation from being used where the tangent is undefined.
To see why the denominator condition matters, use the cosine angle-sum formula:
[ \cos(A+B)=\cos A\cos B-\sin A\sin B. ]
When (\cos A) and (\cos B) are nonzero, factor out their product:
[ \cos(A+B)=\cos A\cos B(1-\tan A\tan B). ]
So if (1-\tan A\tan B=0), then (\cos(A+B)=0). The left side, (\tan(A+B)), is undefined. The right side is also undefined because it divides by zero.
A common mistake is to write
[ \tan(A+B)=\tan A+\tan B. ]
That is generally false. The denominator (1-\tan A\tan B) is part of the sum identity and cannot be dropped.
Other errors to watch for include:
- Using the sum formula for (A-B) without changing the signs.
- Dividing by a cosine before confirming it is nonzero.
- Treating a zero denominator as a value instead of a warning that the expression is undefined.
- Assuming the identity works at every angle just because it is a formula.
Try the sum identity with (A=30^\circ) and (B=45^\circ). The known tangent values are (\tan30^\circ=1/\sqrt3) and (\tan45^\circ=1). Then
[ \frac{\tan30^\circ+\tan45^\circ} {1-\tan30^\circ\tan45^\circ} = \frac{1/\sqrt3+1}{1-1/\sqrt3} =2+\sqrt3. ]
This matches (\tan75^\circ=2+\sqrt3). The denominator is not zero, so the calculation is valid.
Now check (A=B=45^\circ). The denominator becomes (1-(1)(1)=0), while (A+B=90^\circ). Since (\tan90^\circ) is undefined, this is not a case where the formula gives a number. It is a useful domain check, not a failed identity.
Applications: Exact Values, Equation Solving, and Further Trigonometry
The tangent identity is useful when you know the tangents of two angles but need the tangent of their sum or difference. It can also help solve equations or rewrite a trigonometric expression. In every use, identify the angles and check the denominator before simplifying.
Worked cases: Use the formula, then check
For an exact value such as (\tan75^\circ), choose angles that add to (75^\circ) and have known tangent values: (30^\circ+45^\circ). Applying the sum identity gives (2+\sqrt3), without needing a calculator. The denominator is (1-1/\sqrt3), which is not zero.
For a difference, take (A=45^\circ) and (B=30^\circ):
[ \tan15^\circ= \frac{\tan45^\circ-\tan30^\circ} {1+\tan45^\circ\tan30^\circ} = \frac{1-1/\sqrt3}{1+1/\sqrt3} =2-\sqrt3. ]
Here, the denominator is nonzero, and (45^\circ-30^\circ=15^\circ), where tangent exists.
The formula can also help solve a tangent equation. Suppose (\tan(A+B)=r), where (r) is a known value. If (\tan A) and (\tan B) exist, and the denominator is nonzero, then
[ r=\frac{\tan A+\tan B}{1-\tan A\tan B}. ]
You can multiply both sides by the denominator and solve the resulting equation, but retain the restriction (1-\tan A\tan B\ne0). A value that makes this denominator zero is not a valid solution to the original tangent equation.
| Goal | Choose | Result | Check |
|---|---|---|---|
| Find (\tan75^\circ) | (30^\circ+45^\circ) | (2+\sqrt3) | Denominator is nonzero |
| Find (\tan15^\circ) | (45^\circ-30^\circ) | (2-\sqrt3) | Difference denominator is nonzero |
| Check (\tan90^\circ) | (45^\circ+45^\circ) | Undefined | Sum-formula denominator is zero |
A reliable workflow is: identify whether the problem uses a sum or difference; write the matching formula; substitute known tangent values; simplify; and check that all required denominators are nonzero. This method works in degrees or radians, as long as the angle measure stays consistent.
Conclusion: A Reliable Way to Use Tangent Identities
The tangent sum identity comes from the sine and cosine angle-sum formulas. Its quotient form is valid only where the tangents exist and the denominator is nonzero. Remembering both the derivation and the domain check makes it easier to use the formula correctly in exact-value problems and trigonometric equations.
When I help learners review a proof, I suggest they mark each division before moving on. Ask: “What am I dividing by, and can it be zero here?” That small habit can catch a domain error before it spreads through the rest of the work.
FAQ: Tangent Identity Questions
These short answers review the formula, its proof, and its restrictions. The key ideas are that tangent means sine divided by cosine, angle-sum formulas produce the tangent identities, and every denominator must be nonzero. Use the examples below to check common questions about signs, radians, and undefined cases.
What is the tangent sum identity?
(\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}), when both sides are defined.
What is the tangent difference identity?
(\tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}), when both sides are defined.
How do you prove the tangent addition formula?
Write (\tan(A+B)) as (\sin(A+B)/\cos(A+B)), use the sine and cosine angle-sum formulas, then divide by (\cos A\cos B).
Why is the sum denominator (1-\tan A\tan B)?
The cosine sum formula contains (\cos A\cos B-\sin A\sin B). Dividing by (\cos A\cos B) gives (1-\tan A\tan B).
Can I add tangent values directly?
No. In general, (\tan(A+B)) is not (\tan A+\tan B). The correct formula includes the denominator (1-\tan A\tan B).
When is tangent undefined?
Tangent is undefined wherever cosine is zero, because (\tan x=\sin x/\cos x).
What does a zero denominator in the sum formula mean?
If the tangent terms exist and (1-\tan A\tan B=0), then (\cos(A+B)=0). Thus (\tan(A+B)) is undefined.
Does the identity work in radians?
Yes. It works in degrees or radians. Keep the angle units consistent throughout the calculation.
(This article was written by one of our staff writers, Nathan Sterling. Visit our Meet the Team page.)
