Sum and Difference Identities Calculator

Calculate sin(A±B), cos(A±B), and tan(A±B) with step-by-step formula application.

Select Identity & Enter Angles

Result

sin(A + B)
--
sin(A)--
cos(A)--
sin(B)--
cos(B)--
tan(A)--
tan(B)--

All Identities for A = 45° and B = 30°

Step-by-Step Solution

Sum and Difference Identities

The sum and difference identities are fundamental trigonometric formulas that express the sine, cosine, and tangent of the sum or difference of two angles in terms of the trigonometric functions of those angles individually.

The Six Identities

sin(A + B)

Sine of a sum equals the sum of cross products.

sinA cosB + cosA sinB

sin(A - B)

Sine of a difference uses subtraction of cross products.

sinA cosB - cosA sinB

cos(A + B)

Cosine of a sum uses product subtraction.

cosA cosB - sinA sinB

cos(A - B)

Cosine of a difference uses product addition.

cosA cosB + sinA sinB

tan(A + B)

Tangent of a sum expressed as a fraction.

(tanA + tanB) / (1 - tanA tanB)

tan(A - B)

Tangent of a difference expressed as a fraction.

(tanA - tanB) / (1 + tanA tanB)

Applications

  • Finding exact values of trigonometric functions for non-standard angles (e.g., sin(75°) = sin(45° + 30°)).
  • Simplifying trigonometric expressions in calculus and physics.
  • Deriving double-angle and half-angle formulas.
  • Solving trigonometric equations.
  • Signal processing and wave interference calculations.

Tips

Remember the sign patterns: for sine, the sign inside matches the sign between terms. For cosine, the sign inside is opposite to the sign between terms. For tangent, the numerator sign matches while the denominator sign is opposite.