Sum and Difference Identities Notes

Published on: September 29, 2026

This Sum and Difference Identities Calculator works out the sine, cosine or tangent of two angles added together or subtracted, and shows the working line by line. The identity splits the combined angle into values of each angle on its own, which is what makes the result reachable from values that are already familiar. Each value is then replaced by its number and the terms are combined. Following the same pattern every time keeps the signs in the identity straight.

Step-by-step method

  1. Set up the problem.
  2. Write the identity for the chosen function and operation.
  3. Substitute the two angles into the identity.
  4. Replace each trigonometric value with its number.
  5. Work out the products inside the identity.
  6. Simplify to get the value.

Formulas:

\(\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)\)
\(\sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta)\)
\(\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)\)
\(\cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)\)
\(\tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha)\tan(\beta)}\)
\(\tan(\alpha - \beta) = \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)}\)
See Example 1 Hide Example 1

Problem

\(\cos(\alpha + \beta),\; \alpha = 45^{\circ},\; \beta = 30^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the identity for the chosen function and operation.
  3. Substitute the two angles into the identity.
  4. Replace each trigonometric value with its number.
  5. Work out the products inside the identity.
  6. Simplify to get the value.

Step 1

Step 1 - Set up the problem.

In this problem: The sum of the two angles is needed, and they are \(\alpha = 45^{\circ}\) and \(\beta = 30^{\circ}\).

\(\alpha = 45^{\circ},\; \beta = 30^{\circ}\)

Step 2

Step 2 - Write the identity for the chosen function and operation.

In this problem: The cosine of a sum splits into values of the two angles on their own, giving \(\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)\).

\(\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)\)

Step 3

Step 3 - Substitute the two angles into the identity.

In this problem: Replacing \(\alpha\) with \(45^{\circ}\) and \(\beta\) with \(30^{\circ}\) gives \(\cos(45^{\circ})\cos(30^{\circ}) - \sin(45^{\circ})\sin(30^{\circ})\).

\(\cos(\alpha + \beta) = \cos(45^{\circ})\cos(30^{\circ}) - \sin(45^{\circ})\sin(30^{\circ})\)

Step 4

Step 4 - Replace each trigonometric value with its number.

In this problem: Each trigonometric value is worked out on its own, which turns the identity into \(0.71 \times 0.87 - 0.71 \times 0.5\).

\(\cos(\alpha + \beta) = 0.71 \times 0.87 - 0.71 \times 0.5\)

Step 5

Step 5 - Work out the products inside the identity.

In this problem: Multiplying inside each term gives \(0.61\) and \(0.35\).

\(\cos(\alpha + \beta) = 0.61 - 0.35\)

Step 6

Step 6 - Simplify to get the value.

In this problem: Subtracting the two terms gives \(0.26\).

\(\cos(\alpha + \beta) \approx 0.26\)

Final Answer

\(\cos(\alpha + \beta) \approx 0.26\)
See Example 2 Hide Example 2

Problem

\(\sin(\alpha - \beta),\; \alpha = 90^{\circ},\; \beta = 60^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the identity for the chosen function and operation.
  3. Substitute the two angles into the identity.
  4. Replace each trigonometric value with its number.
  5. Work out the products inside the identity.
  6. Simplify to get the value.

Step 1

Step 1 - Set up the problem.

In this problem: The difference of the two angles is needed, and they are \(\alpha = 90^{\circ}\) and \(\beta = 60^{\circ}\).

\(\alpha = 90^{\circ},\; \beta = 60^{\circ}\)

Step 2

Step 2 - Write the identity for the chosen function and operation.

In this problem: The sine of a difference splits into values of the two angles on their own, giving \(\sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta)\).

\(\sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta)\)

Step 3

Step 3 - Substitute the two angles into the identity.

In this problem: Replacing \(\alpha\) with \(90^{\circ}\) and \(\beta\) with \(60^{\circ}\) gives \(\sin(90^{\circ})\cos(60^{\circ}) - \cos(90^{\circ})\sin(60^{\circ})\).

\(\sin(\alpha - \beta) = \sin(90^{\circ})\cos(60^{\circ}) - \cos(90^{\circ})\sin(60^{\circ})\)

Step 4

Step 4 - Replace each trigonometric value with its number.

In this problem: Each trigonometric value is worked out on its own, which turns the identity into \(1 \times 0.5 - 0 \times 0.87\).

\(\sin(\alpha - \beta) = 1 \times 0.5 - 0 \times 0.87\)

Step 5

Step 5 - Work out the products inside the identity.

In this problem: Multiplying inside each term gives \(0.5\) and \(0\).

\(\sin(\alpha - \beta) = 0.5 - 0\)

Step 6

Step 6 - Simplify to get the value.

In this problem: Subtracting the two terms gives \(0.5\).

\(\sin(\alpha - \beta) = 0.5\)

Final Answer

\(\sin(\alpha - \beta) = 0.5\)
See Example 3 Hide Example 3

Problem

\(\sin(\alpha + \beta),\; \alpha = 45^{\circ},\; \beta = 30^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the identity for the chosen function and operation.
  3. Substitute the two angles into the identity.
  4. Replace each trigonometric value with its number.
  5. Work out the products inside the identity.
  6. Simplify to get the value.

Step 1

Step 1 - Set up the problem.

In this problem: The sum of the two angles is needed, and they are \(\alpha = 45^{\circ}\) and \(\beta = 30^{\circ}\).

\(\alpha = 45^{\circ},\; \beta = 30^{\circ}\)

Step 2

Step 2 - Write the identity for the chosen function and operation.

In this problem: The sine of a sum splits into values of the two angles on their own, giving \(\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)\).

\(\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)\)

Step 3

Step 3 - Substitute the two angles into the identity.

In this problem: Replacing \(\alpha\) with \(45^{\circ}\) and \(\beta\) with \(30^{\circ}\) gives \(\sin(45^{\circ})\cos(30^{\circ}) + \cos(45^{\circ})\sin(30^{\circ})\).

\(\sin(\alpha + \beta) = \sin(45^{\circ})\cos(30^{\circ}) + \cos(45^{\circ})\sin(30^{\circ})\)

Step 4

Step 4 - Replace each trigonometric value with its number.

In this problem: Each trigonometric value is worked out on its own, which turns the identity into \(0.71 \times 0.87 + 0.71 \times 0.5\).

\(\sin(\alpha + \beta) = 0.71 \times 0.87 + 0.71 \times 0.5\)

Step 5

Step 5 - Work out the products inside the identity.

In this problem: Multiplying inside each term gives \(0.61\) and \(0.35\).

\(\sin(\alpha + \beta) = 0.61 + 0.35\)

Step 6

Step 6 - Simplify to get the value.

In this problem: Adding the two terms gives \(0.97\).

\(\sin(\alpha + \beta) \approx 0.97\)

Final Answer

\(\sin(\alpha + \beta) \approx 0.97\)
See Example 4 Hide Example 4

Problem

\(\cos(\alpha - \beta),\; \alpha = 60^{\circ},\; \beta = 45^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the identity for the chosen function and operation.
  3. Substitute the two angles into the identity.
  4. Replace each trigonometric value with its number.
  5. Work out the products inside the identity.
  6. Simplify to get the value.

Step 1

Step 1 - Set up the problem.

In this problem: The difference of the two angles is needed, and they are \(\alpha = 60^{\circ}\) and \(\beta = 45^{\circ}\).

\(\alpha = 60^{\circ},\; \beta = 45^{\circ}\)

Step 2

Step 2 - Write the identity for the chosen function and operation.

In this problem: The cosine of a difference splits into values of the two angles on their own, giving \(\cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)\).

\(\cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)\)

Step 3

Step 3 - Substitute the two angles into the identity.

In this problem: Replacing \(\alpha\) with \(60^{\circ}\) and \(\beta\) with \(45^{\circ}\) gives \(\cos(60^{\circ})\cos(45^{\circ}) + \sin(60^{\circ})\sin(45^{\circ})\).

\(\cos(\alpha - \beta) = \cos(60^{\circ})\cos(45^{\circ}) + \sin(60^{\circ})\sin(45^{\circ})\)

Step 4

Step 4 - Replace each trigonometric value with its number.

In this problem: Each trigonometric value is worked out on its own, which turns the identity into \(0.5 \times 0.71 + 0.87 \times 0.71\).

\(\cos(\alpha - \beta) = 0.5 \times 0.71 + 0.87 \times 0.71\)

Step 5

Step 5 - Work out the products inside the identity.

In this problem: Multiplying inside each term gives \(0.35\) and \(0.61\).

\(\cos(\alpha - \beta) = 0.35 + 0.61\)

Step 6

Step 6 - Simplify to get the value.

In this problem: Adding the two terms gives \(0.97\).

\(\cos(\alpha - \beta) \approx 0.97\)

Final Answer

\(\cos(\alpha - \beta) \approx 0.97\)