Sum & Difference Identities Calculator — sin, cos, tan of A±B
Enter two angles A and B in degrees. The calculator applies the six sum and difference identities — sin(A±B), cos(A±B), and tan(A±B) — showing the numeric values and the full substitution for each formula.
°
°
sin(A) = sin(45°)
cos(A) = cos(45°)
sin(B) = sin(30°)
cos(B) = cos(30°)
sin(A+B) = sinA·cosB + cosA·sinB
cos(A+B) = cosA·cosB − sinA·sinB
tan(A+B) = (tanA + tanB) / (1 − tanA·tanB)
sin(A−B) = sinA·cosB − cosA·sinB
cos(A−B) = cosA·cosB + sinA·sinB
tan(A−B) = (tanA − tanB) / (1 + tanA·tanB)
How does this calculator work?
sin(A±B) = sinA cosB ± cosA sinB; cos(A±B) = cosA cosB ∓ sinA sinB; tan(A±B) = (tanA±tanB)/(1∓tanA tanB). These exact identities work for any angles. They are used to find exact trig values, derive double-angle and half-angle formulas, and simplify calculus expressions.
Formula
How this is calculated
The sum and difference identities express the sine, cosine, and tangent of the sum or difference of two angles entirely in terms of sines and cosines of the original angles. They are fundamental identities in trigonometry — not approximations — true for any angles A and B.
For sine: sin(A+B) = sin A cos B + cos A sin B and sin(A−B) = sin A cos B − cos A sin B. The signs switch because you are rotating in the opposite direction. For cosine the sign rule is reversed: cos(A+B) = cos A cos B − sin A sin B (note the minus) and cos(A−B) = cos A cos B + sin A sin B. For tangent, dividing sin(A±B) by cos(A±B) and simplifying yields tan(A±B) = (tan A ± tan B) / (1 ∓ tan A tan B). Tangent is undefined when the denominator equals zero, which occurs at certain angle combinations (e.g. A = 45° and B = 45° for the sum, because tan 90° is undefined).
These identities are used to derive other identities (double-angle, half-angle, product-to-sum), to expand expressions in calculus and Fourier analysis, and to compute exact values of non-standard angles — for example sin(75°) = sin(45°+30°) = (√6+√2)/4.
Frequently asked questions
It follows from the geometry of rotation: when you combine two rotations, the cosine of the total angle involves a subtraction because the x-component (cosine) of the combined rotation picks up a cross-term with a negative sign. This is easiest to see by deriving the identity from the rotation matrix or from Euler's formula.
tan(A+B) is undefined whenever its denominator 1 − tan A·tan B = 0, i.e. when tan A·tan B = 1. This means A+B is an odd multiple of 90° (where the combined angle lands on a vertical line). For example A = 30°, B = 60° gives tan 30°·tan 60° = (1/√3)·√3 = 1, making tan(90°) undefined.
The identities are equally valid in radians — the formulas do not depend on the unit. This calculator accepts degrees for convenience; to use radians, convert first: radians = degrees × π/180.
Also known as
TG we-Calculate Editorial Team. (2026). Sum & Difference Identities Calculator — sin, cos, tan of A±B [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/sum-difference-identities-calculator
TG we-Calculate Editorial Team. "Sum & Difference Identities Calculator — sin, cos, tan of A±B." TG we-Calculate. 2026. https://we-calculate.com/calculator/sum-difference-identities-calculator.
TG we-Calculate Editorial Team, "Sum & Difference Identities Calculator — sin, cos, tan of A±B," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/sum-difference-identities-calculator
@misc{wecalculate_sum_difference_identities_calculator, title = {Sum & Difference Identities Calculator — sin, cos, tan of A±B}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/sum-difference-identities-calculator}}, year = {2026}, note = {TG we-Calculate} }
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