Sum Difference Identities Calculator - Sum and Difference Identity Solver

A sum difference identities calculator is a trigonometry tool that takes two angles, alpha and beta, and returns the sine, cosine, and tangent of (alpha + beta) and (alpha - beta) using the standard angle addition and subtraction formulas. The six outputs sit on one screen, so a homework problem that asks for sin(75), cos(75), and tan(75) is one input away, in degrees or radians.

• • Evaluating sin(75) and cos(75) for a 30 + 45 problem: Enter alpha = 30, beta = 45, and read sin(75) and cos(75) from the sum row.
• • Verifying a hand-written cos(alpha - beta) expansion: Expand cos(alpha - beta) on paper, then enter the same alpha and beta to confirm.
• • Working through tangent asymptotes: Pick alpha and beta so alpha + beta lands on 90 degrees and see tan(alpha + beta) flip to 'undefined'.
• • Switching between degrees and radians: Toggle the unit selector so the same identities work in either unit.

For the alpha = beta special case, the Double Angle Formula Calculator returns sin(2 theta), cos(2 theta), and tan(2 theta) directly from the same identity framework.

The solver reads alpha, beta, and the unit selector, converts both angles to a single radian representation, builds the two composite angles (alpha + beta) and (alpha - beta), and evaluates the six identities on those two angles. A single input drives the full answer.

• alpha: The first input angle, converted to radians internally.
• beta: The second input angle, also converted to radians internally.
• alpha + beta: The composite angle that drives the sin/cos/tan of the sum row.
• alpha - beta: The composite angle that drives the sin/cos/tan of the difference row.
• Identity check (sum): Right-hand side of the sine sum identity, shown in the identity checks row at the bottom of the results panel.
• Identity check (difference): Right-hand side of the cosine difference identity, shown in the identity checks row at the bottom of the results panel.

The identity check on the results panel re-evaluates the right-hand side of the sine sum and cosine difference identities separately. When the two numbers match to four decimal places, the formulas are behaving as predicted.

According to Wolfram MathWorld: Addition Formulas, the six angle sum and difference identities are sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β), cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β), and tan(α ± β) = (tan(α) ± tan(β)) / (1 ∓ tan(α)tan(β)).

According to Cuemath: Sum and Difference Identities, sin(75°) = (sqrt(6) + sqrt(2)) / 4 ≈ 0.9659, which is exactly what the sum identity produces for alpha = 30° and beta = 45°.

Once the sum and difference identities feel comfortable, the natural next step is the special case alpha = beta, which the Double Angle Identities Calculator covers for sin(2 theta), cos(2 theta), and tan(2 theta) in a single step.

Four concepts make the identities on this page intuitive. Once they are clear the six identities fall out as one-line applications.

Euler's formula is the most compact way to remember the identities: multiplying cos(theta) + i sin(theta) for the two angles gives cos(alpha + beta) + i sin(alpha + beta), and matching real and imaginary parts recovers the four sine and cosine identities in two lines.

Going the other direction, the Half Angle Calculator applies the half-angle identity to recover sin(theta/2), cos(theta/2), and tan(theta/2) from the same sin(theta) and cos(theta) values that drive the sum and difference identities on this page.

Using the solver takes a few seconds. Pick the unit, type alpha and beta, and read the six identity outputs plus the two identity checks from the results panel.

1. 1 Pick the unit: Choose degrees or radians. The solver does the conversion internally.
2. 2 Enter alpha: Type the first angle. The default of 30 degrees is the most common reference value, but any real number works, including negatives and angles outside 0 to 360.
3. 3 Enter beta: Type the second angle. The default of 45 pairs well with alpha = 30 to give the clean sin(75) and cos(75) values from the textbook.
4. 4 Read the sum row: The first three results show sin(alpha + beta), cos(alpha + beta), and tan(alpha + beta). tan(alpha + beta) shows 'undefined' when cos(alpha + beta) is effectively zero.
5. 5 Read the difference row: The next three results show sin(alpha - beta), cos(alpha - beta), and tan(alpha - beta).
6. 6 Verify with the identity checks: The two extra rows re-evaluate the right-hand side of the sine sum identity and the cosine difference identity.

Suppose alpha = 22.5 degrees and beta = 67.5 degrees. Enter 22.5 and 67.5 with degrees selected, then read sin(90) = 1, cos(90) = 0 from the sum row and sin(-45) = -0.7071, cos(-45) = 0.7071 from the difference row.

If a problem gives the inputs in degrees but the textbook identity table uses radians, the Angle Converter is the fastest way to switch the unit on a known angle before re-entering the value here.

Putting the six identity outputs and two identity checks on one screen collapses a multi-step manual problem into a single read.

• • All six identity outputs in one view: sin, cos, and tan for (alpha + beta) and (alpha - beta) appear together.
• • Degree and radian support side by side: Switching the unit selector re-evaluates the same six outputs in the new unit.
• • Built-in identity check: The right-hand side of the sine sum and cosine difference identities are re-evaluated in the identity checks row at the bottom of the results panel.
• • Tangent asymptotes handled cleanly: The solver reports 'undefined' when the composite cosine is effectively zero.
• • Negative and out-of-range inputs work: alpha and beta can be negative, larger than 360, or both.
• • Faster than hand expansion: Once alpha + beta or alpha - beta is known, the identity outputs are immediate.

For problems that ask for cos(2 theta) in particular, the Cos 2 Theta Calculator returns the same cosine value alongside the alternative forms cos squared - sin squared and 1 - 2 sin squared.

Most of the time the solver returns the expected six values, but a handful of factors control what the tool can and cannot do.

• • The solver returns the principal real values; it does not compute complex-valued trig outputs.
• • Floating-point arithmetic means the identity checks match to roughly 15 significant digits.

According to Wikipedia: List of trigonometric identities, the angle sum and difference identities form the foundation that all other derived identities follow from.

If a factor keeps forcing alpha + beta or alpha - beta past 360 degrees or into negative territory, the Coterminal Angle Calculator reduces any real angle to its principal value, which is often the cleanest way to spot the asymptote threshold before re-entering the values here.