Sum and Difference Formulas — Sin, Cos, Tan

Trigonometry

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.