Sum and Difference Identities Calculator

This sum and difference identities calculator evaluates the sine, cosine and tangent of the sum or difference of two angles, using the trigonometric addition formulas. Enter two angles A and B, choose degrees or radians, pick whether you want the sum A plus B or the difference A minus B, and the calculator returns sin, cos and tan of the combined angle. The addition formulas let you expand sin(A plus B) and the others in terms of the sines and cosines of A and B alone, which is the basis for many proofs, for simplifying expressions and for working with exact values such as sin 75 degrees. The calculator both evaluates the combined angle directly and lets you confirm the identities. The formulas, a worked example and the assumptions are set out below.

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Angle A

Angle B

Angle unit

Operation

sin = 0.965926

cos(result)0.258819

tan(result)3.732051

The formula

The addition formulas are: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B −/+ sin A sin B; and tan(A ± B) = (tan A ± tan B) / (1 −/+ tan A tan B). The calculator works in radians internally and shows the result for your chosen operation.

Worked example

sin(30 + 45) = sin 30 cos 45 + cos 30 sin 45 = 0.5(0.707107) + 0.866025(0.707107), about 0.965926, which is sin 75 degrees. Enter 30 and 45 with sum selected to confirm.

Frequently asked questions

What are the sum and difference identities?

They expand a function of A plus or minus B: sin(A+B) = sin A cos B + cos A sin B, cos(A+B) = cos A cos B - sin A sin B, and the matching tan formula.

Why are they useful?

They give exact values for angles like 15 or 75 degrees, simplify expressions, and are used throughout calculus and physics.

Degrees or radians?

Either. Pick the unit in the dropdown. The calculator converts to radians to evaluate.

Who this calculator is for

This calculator is for trigonometry and calculus students working with the addition formulas.