# Sum and Difference Formulas — Sin, Cos, Tan

[Trigonometry](/content/tag/trigonometry/index.html)

TL;DR

The sum and difference formulas of trigonometry give the sine, cosine, and tangent of A±B in terms of the trig functions of A and B separately — six identities that let you compute exact values for non-standard angles like 15° or 75°. This article gives the six formulas, the unit-circle proof, three worked examples in degrees and radians, the sign-flip mnemonic for cosine, and the common mistakes around tangent's denominator.

## Six Identities That Reduce Any Angle Sum to Familiar Pieces

The **sum and difference formulas** — also called **sum and difference identities** — give:

- sin⁡(A±B) in terms of sin⁡A,cos⁡A,sin⁡B,cos⁡B.
- cos⁡(A±B) in terms of sin⁡A,cos⁡A,sin⁡B,cos⁡B.
- tan⁡(A±B) in terms of tan⁡A,tan⁡B.

These six identities bridge "I know sine and cosine of 30°, 45°, 60°" and "I can compute sine of 15° = 45°−30° exactly."

## The Six Formulas of Sum and Difference

1.   sin(A + B) = sin A cos B + cos A sin B  
2.   sin(A - B) = sin A cos B - cos A sin B  
3.   cos(A + B) = cos A cos B - sin A sin B  
4.   cos(A - B) = cos A cos B + sin A sin B  
5.   tan(A + B) = (tan A + tan B) / (1 - tan A tan B)  
6.   tan(A - B) = (tan A - tan B) / (1 + tan A tan B)

The cosine identities carry an **opposite-sign rule**: the sign between the two terms is the _opposite_ of the sign between the angles. The sine and tangent identities keep the same sign as the angle operation.

### Quick Facts
- Holds for all real A and B — no domain restriction beyond where tan is defined.
- Tangent identity caveat: tan(A+B) is undefined when 1−tanAtanB=0, i.e., when tanAtanB=1.
- In radians: every angle in these formulas is in radians by default for calculus contexts; degrees work identically for non-calculus contexts.
- Reduces to double-angle identities when A=B: sin(2A)=2sinAcosA, cos(2A)=cos²A−sin²A.

## Three Worked Examples of Sum and Difference

1. **Quick**. Compute sin⁡75° exactly.
   - Write 75°=45°+30°, apply the sine sum formula:  
sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°).
   - Substitute special-angle values: sin(45°)=cos(45°)=√2/2, sin(30°)=1/2, cos(30°)=√3/2.
   - Final answer: sin(75°) = (√6 + √2) / 4.

2. **Standard**. Compute cos⁡15° exactly using cos(45°−30°).
   - Student writes: cos(45°−30°)=cos(45°)cos(30°)+sin(45°)sin(30°).
   - Substitute and get: cos(15°) = (√6 + √2) / 4.

3. **Stretch**. Compute tan⁡105° exactly.
   - Write 105°=60°+45°, apply the tangent sum formula:  
tan(105°) = (tan(60°) + tan(45°)) / (1 - tan(60°)tan(45°).
   - Final answer: tan(105°) = -2 - √3 ≈ -3.732.

## Key Takeaways
- The **sum and difference formulas** are six identities expressing sin(A±B), cos(A±B), and tan(A±B) in terms of trig functions of A and B alone.
- The **cosine identities flip signs**: cos(A+B)=cosAcosB−sinAsinB, cos(A−B)=cosAcosB+sinAsinB.
- The **tangent identities have the same flip in the denominator** — the signs differ based on angle operations.
- Historical context includes Ptolemy's theorem and contributions from mathematicians like Hipparchus and Bhaskara II.