Sum and Difference Identities Formula

Sum and difference identities are formulas that express (A B), (A B), and (A B) in terms of A, A, B, and B.

The Formula

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B
cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B
tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}

When to use: What happens when you combine two rotations? If you rotate by angle AA and then by angle BB, the result involves both angles interacting. The sum and difference formulas tell you exactly how the trig values of two separate angles combine. They're like a multiplication rule for rotations—the result isn't simply adding the trig values, but mixing sines and cosines together.

Quick Example

cos⁡(75°)=cos⁡(45°+30°)=cos⁡45°cos⁡30°−sin⁡45°sin⁡30°\cos(75°) = \cos(45° + 30°) = \cos 45°\cos 30° - \sin 45°\sin 30° =22⋅32−22⋅12=6−24= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}

Notation

Note the sign pattern: in the cosine formula, ±\pm becomes ∓\mp (signs are opposite). In sine and tangent, signs match.

What This Formula Means

Formulas that express sin⁡(A±B)\sin(A \pm B), cos⁡(A±B)\cos(A \pm B), and tan⁡(A±B)\tan(A \pm B) in terms of sin⁡A\sin A, cos⁡A\cos A, sin⁡B\sin B, and cos⁡B\cos B.

What happens when you combine two rotations? If you rotate by angle AA and then by angle BB, the result involves both angles interacting. The sum and difference formulas tell you exactly how the trig values of two separate angles combine. They're like a multiplication rule for rotations—the result isn't simply adding the trig values, but mixing sines and cosines together.

Formal View

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B; cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B; tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}

Worked Examples

Example 1

easy
Find the exact value of cos⁡(75°)\cos(75°) using the sum identity.

Answer

cos⁡(75°)=6−24\cos(75°) = \frac{\sqrt{6} - \sqrt{2}}{4}

First step

1
Write 75°=45°+30°75° = 45° + 30°.

Full solution

  1. 2
    Apply the cosine sum formula: cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A+B) = \cos A \cos B - \sin A \sin B.
  2. 3
    Substitute: cos⁡(75°)=cos⁡(45°)cos⁡(30°)−sin⁡(45°)sin⁡(30°)\cos(75°) = \cos(45°)\cos(30°) - \sin(45°)\sin(30°).
  3. 4
    =22⋅32−22⋅12=64−24=6−24= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}.
The sum and difference identities let us find exact values for angles that are sums or differences of standard angles (30°, 45°, 60°). Here we decomposed 75° as 45° + 30° and applied the cosine addition formula.

Example 2

medium
Simplify sin⁡(x+y)+sin⁡(x−y)\sin(x + y) + \sin(x - y).

Example 3

medium
Show that sin⁡(x+π/2)=cos⁡x\sin(x + \pi/2) = \cos x.

Common Mistakes

  • Splitting cos⁡(A+B)\cos(A+B) as cos⁡A+cos⁡B\cos A+\cos B - it expands to cos⁡Acos⁡B−sin⁡Asin⁡B\cos A\cos B-\sin A\sin B, a product mix.
  • Getting the cosine sign wrong - in cosine the inside ++ becomes a −- between the products (∓\mp flips).
  • Using matching signs for cosine - sine and tangent keep the sign, cosine reverses it.

Why This Formula Matters

They unlock exact values for angles outside the special set and are the launchpad for double-angle and product-to-sum work. The fatal error — assuming sin⁡(A+B)=sin⁡A+sin⁡B\sin(A+B)=\sin A+\sin B — wrecks all of trig, since trig functions are not linear. Recognizing it by "Is the argument of the trig function a sum or difference of two angles I want to break apart?" — rather than by familiar numbers — is what lets a student tell it apart from double-angle identities and pythagorean identities and distributing the function (wrong) in a mixed problem set.

Frequently Asked Questions

What is the Sum and Difference Identities formula?

Formulas that express sin⁡(A±B)\sin(A \pm B), cos⁡(A±B)\cos(A \pm B), and tan⁡(A±B)\tan(A \pm B) in terms of sin⁡A\sin A, cos⁡A\cos A, sin⁡B\sin B, and cos⁡B\cos B.

How do you use the Sum and Difference Identities formula?

What happens when you combine two rotations? If you rotate by angle AA and then by angle BB, the result involves both angles interacting. The sum and difference formulas tell you exactly how the trig values of two separate angles combine. They're like a multiplication rule for rotations—the result isn't simply adding the trig values, but mixing sines and cosines together.

What do the symbols mean in the Sum and Difference Identities formula?

Note the sign pattern: in the cosine formula, ±\pm becomes ∓\mp (signs are opposite). In sine and tangent, signs match.

Why is the Sum and Difference Identities formula important in Math?

They unlock exact values for angles outside the special set and are the launchpad for double-angle and product-to-sum work. The fatal error — assuming sin⁡(A+B)=sin⁡A+sin⁡B\sin(A+B)=\sin A+\sin B — wrecks all of trig, since trig functions are not linear. Recognizing it by "Is the argument of the trig function a sum or difference of two angles I want to break apart?" — rather than by familiar numbers — is what lets a student tell it apart from double-angle identities and pythagorean identities and distributing the function (wrong) in a mixed problem set.

What do students get wrong about Sum and Difference Identities?

The procedure for sum and difference identities is the easy part; the trap is splitting cos⁡(A+B)\cos(A+B) as cos⁡A+cos⁡B\cos A+\cos B. Asking "Is the argument of the trig function a sum or difference of two angles I want to break apart?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Sum and Difference Identities formula?

Before studying the Sum and Difference Identities formula, you should understand: trig identities pythagorean, trigonometric functions.

← Back to Sum and Difference Identities See All Examples Practice Problems