Sum and Difference Formulas Calculator

The angle addition and subtraction formulas let you compute trig values for sums or differences of angles without a calculator. They are how you get exact values for angles the standard tables miss, and they are the engine behind most trig simplification.

sin(A + B) = sin A cos B + cos A sin B sin(A − B) = sin A cos B − cos A sin B cos(A + B) = cos A cos B − sin A sin B cos(A − B) = cos A cos B + sin A sin B tan(A + B) = (tan A + tan B) / (1 − tan A tan B) tan(A − B) = (tan A − tan B) / (1 + tan A tan B)

One pattern to notice: for sine, the sign in the formula matches the sign in the argument. For cosine, the signs are reversed. That reversal trips up students more than anything else.

A classic use: sin(75°) = sin(45° + 30°) = sin45 cos30 + cos45 sin30 = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4. That is an exact value you would not get from the standard 30-60-90 table alone.

These formulas are provable from the unit circle using rotation matrices. Rotate a point by A, then by B, and the combined rotation is A + B. Expanding that matrix product gives exactly the formulas above, which is the cleanest derivation there is.

Tan(A+B) is undefined when the denominator equals zero, which happens when tan A × tan B = 1. A = B = 45° is the case everyone meets first: both tangents are 1, the denominator is 1 − 1 = 0, and A + B = 90° where tangent genuinely has an asymptote. The calculator tests that denominator directly and says “undefined” rather than printing an enormous number.

The result panel shows each formula with your own values substituted in, and then evaluates the same angle directly for comparison. Those two routes are independent, so watching them agree is a real check on the identity rather than a restatement of it.