Sum/Difference Identities — Formulas, Table & Examples

Sum/Difference Identities

Trig identities which show how to find the sine, cosine, or tangent of the sum or difference of two given angles.

Sum/Difference Identities

Two tangent sum/difference identities: tan(x+y)=(tan x+tan y)/(1−tan x tan y); tan(x−y)=(tan x−tan y)/(1+tan x tan y)

Key Formula

\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B

\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B

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

Where:

$A$ = The first angle
$B$ = The second angle
$\pm / \mp$ = The upper signs go together (sum) and the lower signs go together (difference). For cosine, the sign in the result is opposite to the sign in the argument.

Worked Example

Problem: Find the exact value of sin(75°).

Step 1: Rewrite 75° as the sum of two familiar angles.

75° = 45° + 30°

Step 2: Apply the sine sum identity: sin(A + B) = sin A cos B + cos A sin B.

\sin(75°) = \sin(45°)\cos(30°) + \cos(45°)\sin(30°)

Step 3: Substitute the known exact values for each trig function.

= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}

Step 4: Multiply the terms.

= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}

Step 5: Combine over a common denominator.

= \frac{\sqrt{6} + \sqrt{2}}{4}

Answer: sin(75°) = (√6 + √2) / 4 ≈ 0.9659

Another Example

This example uses radians instead of degrees, applies the cosine difference identity (rather than the sine sum identity), and highlights the sign-flip rule unique to cosine. It also shows that sin(75°) = cos(15°), reinforcing the cofunction relationship.

Problem: Find the exact value of cos(π/12).

Step 1: Rewrite π/12 as the difference of two familiar angles. Note that π/12 = 15° = 60° − 45°.

\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}

Step 2: Apply the cosine difference identity: cos(A − B) = cos A cos B + sin A sin B. Notice the plus sign — for cosine, the sign flips relative to the argument.

\cos\!\left(\frac{\pi}{12}\right) = \cos\frac{\pi}{3}\cos\frac{\pi}{4} + \sin\frac{\pi}{3}\sin\frac{\pi}{4}

Step 3: Substitute exact values.

= \frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}

Step 4: Simplify each product and combine.

= \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}

Answer: cos(π/12) = (√6 + √2) / 4 ≈ 0.9659

Frequently Asked Questions

What is the difference between the sum identity and the difference identity?

They are the same formula with opposite signs. For sine, you replace '+' with '−' when the angles are subtracted. For cosine, the sign in the formula is opposite to the sign in the argument: cos(A + B) uses a minus sign between its terms, while cos(A − B) uses a plus sign. The tangent identity follows a similar pattern where both the numerator and denominator signs swap.

When do you use sum and difference identities?

You use them whenever you need the exact value of a trig function at a non-standard angle that can be written as a sum or difference of standard angles (like 30°, 45°, 60°, 90°). They also appear frequently when simplifying trigonometric expressions, proving other identities, and solving trig equations in precalculus and calculus.

How do you remember the sign in the cosine sum/difference identity?

A helpful mnemonic: cosine is 'contrary.' The sign inside the cosine argument and the sign in the expanded formula are opposite. So cos(A + B) has a minus sign between its terms, and cos(A − B) has a plus sign. Sine and tangent, by contrast, keep the same sign.

Sum/Difference Identities vs. Double Angle Identities

	Sum/Difference Identities	Double Angle Identities
Definition	Express trig functions of (A ± B) using trig functions of A and B separately	Express trig functions of 2A using trig functions of A alone
Example formula	sin(A + B) = sin A cos B + cos A sin B	sin(2A) = 2 sin A cos A
Relationship	The general case for any two angles	A special case of the sum identity where B = A
When to use	Finding exact values of non-standard angles (e.g., 75°, 15°, 105°)	Simplifying expressions involving 2θ, or solving equations with mixed single and double angles

Why It Matters

Sum and difference identities are central to precalculus and are tested heavily on the AP Calculus and SAT/ACT exams. In calculus, they are essential for deriving the derivatives of sine and cosine from the limit definition. Engineers and physicists use them to analyze wave interference, signal processing, and alternating current circuits where combining oscillations at different phases is routine.

Common Mistakes

Mistake: Using the wrong sign in the cosine identity — writing cos(A + B) = cos A cos B + sin A sin B instead of the correct minus sign.

Correction: Remember that cosine is 'contrary': the sign in the expanded form is always opposite to the sign in the argument. cos(A + B) = cos A cos B − sin A sin B, and cos(A − B) = cos A cos B + sin A sin B.

Mistake: Distributing trig functions across addition, writing sin(A + B) = sin A + sin B.

Correction: Trig functions are not linear — you cannot distribute them like multiplication. You must use the full identity: sin(A + B) = sin A cos B + cos A sin B. A quick numerical check (e.g., sin(60°) ≠ sin(30°) + sin(30°)) confirms this.

Related Terms

Trig Identities — The broader family these identities belong to
Sine — One of the three functions with a sum/difference identity
Cosine — One of the three functions with a sum/difference identity
Tangent — One of the three functions with a sum/difference identity
Double Angle Identities — Special case where both angles are equal
Half Angle Identities — Derived from sum/difference identities
Angle — The input values A and B in the identities
Unit Circle — Source of exact trig values used in these identities