Sum and Difference Identities Calculator
Enter two angles and choose a trigonometric function to evaluate sin(A+B), sin(A-B), cos(A+B), cos(A-B), tan(A+B), tan(A-B), and their reciprocals cot, sec, and csc. The calculator applies the sum and difference identities, shows every step of the working, and displays the decimal result alongside the exact symbolic form for common angles.
What are sum and difference identities?
Sum and difference identities (also called angle addition and subtraction formulas) express the trigonometric functions of a combined angle A+B or A-B entirely in terms of the individual functions of A and B. For sine, the sum identity reads: sin(A+B) = sinA cosB + cosA sinB. The cosine counterpart is: cos(A+B) = cosA cosB - sinA sinB. The tangent identity is: tan(A+B) = (tanA + tanB) / (1 - tanA tanB). Each formula has a corresponding difference version where the sign of the B terms reverses. These six identities are among the most important results in trigonometry and underpin the derivation of double-angle, half-angle, product-to-sum, and sum-to-product identities.
How to use this calculator
Select degree or radian mode, then enter angles A and B. The calculator evaluates all six trigonometric functions at both A+B and A-B, showing the decimal result and, for sine and cosine at multiples of 15 degrees, the exact symbolic form such as (root6 + root2) / 4. The "show your work" panel traces every arithmetic step of the chosen identity. A practical use is to evaluate a non-standard angle by decomposing it: sin(75 degrees) = sin(45 + 30) gives the exact value (root6 + root2) / 4 without a calculator table.
Derivation and key applications
The cosine difference identity cos(A-B) = cosA cosB + sinA sinB is often derived geometrically using the unit circle and the distance formula. The remaining five identities follow by substitution and the co-function relations (sin x = cos(90 - x)). Applications span all of mathematics and physics: proving other identities (double-angle, half-angle, product-to-sum), simplifying Fourier analysis expressions, evaluating integrals, solving trigonometric equations, and computing phase shifts in electronics and signal processing.
Reciprocal functions: cot, sec, and csc
The cotangent sum identity is cot(A+B) = (cotA cotB - 1) / (cotA + cotB). Secant and cosecant follow directly from the cosine and sine identities: sec(A+B) = 1 / cos(A+B) and csc(A+B) = 1 / sin(A+B), so their expanded forms are the reciprocal of the appropriate primary identity. All six are undefined at specific angle combinations where their denominator or the underlying sine or cosine is zero. The tangent formula is undefined when A + B equals 90 degrees or any odd multiple of 90 degrees, because 1 - tanA tanB vanishes there.
Sum and Difference Identities - All Six Functions
| Function | Sum identity f(A+B) | Difference identity f(A-B) |
|---|---|---|
| sin | sinA cosB + cosA sinB | sinA cosB - cosA sinB |
| cos | cosA cosB - sinA sinB | cosA cosB + sinA sinB |
| tan | (tanA + tanB) / (1 - tanA tanB) | (tanA - tanB) / (1 + tanA tanB) |
| cot | (cotA cotB - 1) / (cotA + cotB) | (cotA cotB + 1) / (cotB - cotA) |
| sec | 1 / (cosA cosB - sinA sinB) | 1 / (cosA cosB + sinA sinB) |
| csc | 1 / (sinA cosB + cosA sinB) | 1 / (sinA cosB - cosA sinB) |
The six fundamental sum and difference identities. Reciprocal identities (cot, sec, csc) are derived from the primary three.
Frequently asked questions
Why do the formulas have mixed signs?
The sign pattern comes from the unit-circle proof. For sine, the sum uses a plus between the two cross-products (sinA cosB + cosA sinB), while the difference swaps to minus. For cosine it is the reverse: the sum subtracts the sine product (cosA cosB - sinA sinB), and the difference adds it. A useful memory aid is that sine and cosine formulas always mix both functions of A and B, while tangent uses only tangent values in a fraction.
How do I find sin(75 degrees) using these identities?
Write 75 degrees as 45 + 30. Then sin(75) = sin(45) cos(30) + cos(45) sin(30) = (root2/2)(root3/2) + (root2/2)(1/2) = root6/4 + root2/4 = (root6 + root2) / 4, which is approximately 0.9659. This technique works for any angle that can be expressed as a sum or difference of the standard special angles: 0, 30, 45, 60, and 90 degrees.
What is the difference between the sum identity and a double-angle identity?
A double-angle identity is just the sum identity applied when both angles are equal (A = B). Setting A = B in sin(A+B) = sinA cosB + cosA sinB immediately gives sin(2A) = 2 sinA cosA. Similarly, cos(2A) = cos^2(A) - sin^2(A) comes from the cosine sum formula. Double-angle identities are a special case, not separate formulas.
When is the tangent formula undefined?
tan(A+B) = (tanA + tanB) / (1 - tanA tanB) is undefined when the denominator 1 - tanA tanB equals zero, which happens when tanA tanB = 1. This occurs when A + B = 90 degrees (or 270, 450, etc.), because tan(90) itself is undefined. It also fails when A or B is itself 90 degrees, since tanA or tanB would be undefined. In those cases, switching to the sine or cosine form, and then dividing, gives the correct limiting value.
Can these identities be used in reverse to simplify expressions?
Yes. If you see an expression of the form sinX cosY + cosX sinY anywhere in a proof or integral, you can collapse it to sin(X+Y). This product-to-sum direction is common in simplifying integrals, Fourier series coefficients, and complex-number computations in electrical engineering. The key is recognising the pattern: two cross-products of sin and cos with a plus or minus sign almost always collapse to a single sine or cosine of a sum or difference.