Sum & Difference Identities: Formulas, Examples & Tips

Sum and difference identities are powerful tools for simplifying trigonometric expressions. Instead of treating an angle such as 75° or 15° as a mystery, you can break it into familiar angles, apply an identity, substitute known values, and simplify.

Sum and Difference Identities: Simplifying Complex Trigonometric Expressions

If you are studying high school trigonometry, Algebra 2, precalculus, or college mathematics, you will eventually encounter trigonometric expressions involving sums and differences of angles. Examples include sin(75°), cos(15°), and tan(105°).

The good news is that these expressions can often be simplified using a small collection of formulas. In this guide, we will learn the identities step by step, work through exact-value examples, discover an easy memory trick, identify common mistakes, and finish with practice problems.

1. The Main Idea

The key idea is simple: break a complicated angle into two familiar angles.

For example, 75° can be written as:

75° = 45° + 30°

Both 45° and 30° have familiar sine and cosine values. The sum identity allows us to turn sin(75°) into an expression involving those familiar values.

The strategy:

1. Rewrite the angle as a sum or difference.

2. Choose the correct identity.

3. Substitute known trig values.

4. Simplify carefully.

5. Check that the final value makes sense.

2. Sine Sum and Difference Identities

There are two important sine identities.

Sine of a Sum

sin(A + B) = sin A × cos B + cos A × sin B

Sine of a Difference

sin(A − B) = sin A × cos B − cos A × sin B

Notice the important pattern: sine keeps the same sign between the two terms. A plus inside the angle gives a plus between the terms, while a minus gives a minus.

3. Cosine Sum and Difference Identities

Cosine uses a slightly different pattern.

Cosine of a Sum

cos(A + B) = cos A × cos B − sin A × sin B

Cosine of a Difference

cos(A − B) = cos A × cos B + sin A × sin B

Here is the pattern students often need to remember: cosine changes the sign. A plus inside the angle produces a minus between the terms, while a minus inside the angle produces a plus.

4. Tangent Sum and Difference Identities

Tangent also has sum and difference identities. They are especially useful when the individual tangent values are known.

Tangent of a Sum

tan(A + B) = (tan A + tan B) ÷ (1 − tan A × tan B)

Tangent of a Difference

tan(A − B) = (tan A − tan B) ÷ (1 + tan A × tan B)

Be careful with the tangent formulas. The signs in the numerator and denominator follow different patterns.

5. Worked Example: Find sin(75°)

Step 1: Break apart the angle

75° = 45° + 30°

Step 2: Use the sine sum identity

sin(75°) = sin(45° + 30°)

sin(75°) = sin 45° × cos 30° + cos 45° × sin 30°

Step 3: Substitute familiar values

sin(75°) = (√2 ÷ 2) × (√3 ÷ 2) + (√2 ÷ 2) × (1 ÷ 2)

Step 4: Simplify

sin(75°) = (√6 + √2) ÷ 4

Final answer: sin(75°) = (√6 + √2) ÷ 4, approximately 0.966.

6. Worked Example: Find cos(15°)

Step 1: Rewrite 15° as a difference of familiar angles.

15° = 45° − 30°

Step 2: Apply the cosine difference identity.

cos(15°) = cos(45° − 30°)

cos(15°) = cos 45° × cos 30° + sin 45° × sin 30°

Step 3: Substitute the known values.

cos(15°) = (√2 ÷ 2) × (√3 ÷ 2) + (√2 ÷ 2) × (1 ÷ 2)

Step 4: Simplify.

cos(15°) = (√6 + √2) ÷ 4

Final answer: cos(15°) = (√6 + √2) ÷ 4, approximately 0.966.

7. Worked Example: Find cos(105°)

Use 105° = 60° + 45°.

cos(105°) = cos(60° + 45°)

Apply the cosine sum identity:

cos(105°) = cos 60° × cos 45° − sin 60° × sin 45°

Substitute the familiar values:

cos(105°) = (1 ÷ 2) × (√2 ÷ 2) − (√3 ÷ 2) × (√2 ÷ 2)

Therefore:

cos(105°) = (√2 − √6) ÷ 4

Final answer: cos(105°) = (√2 − √6) ÷ 4, approximately −0.259.

8. Worked Example: Find tan(75°)

Write 75° as 45° + 30°.

tan(75°) = tan(45° + 30°)

Use the tangent sum identity:

tan(75°) = (tan 45° + tan 30°) ÷ (1 − tan 45° × tan 30°)

Substitute tan 45° = 1 and tan 30° = √3 ÷ 3:

tan(75°) = (1 + √3 ÷ 3) ÷ (1 − √3 ÷ 3)

Multiply the numerator and denominator by 3:

tan(75°) = (3 + √3) ÷ (3 − √3)

Rationalizing the denominator gives:

tan(75°) = 2 + √3

Final answer: tan(75°) = 2 + √3, approximately 3.732.

9. The Best Memory Trick

Remember: Sine stays, cosine changes.

For sine, the sign inside the identity matches the sign between the two terms:

sin(A + B) → +     sin(A − B) → −

For cosine, the sign switches:

cos(A + B) → −     cos(A − B) → +

Memory sentence: “Sine stays the same; cosine changes the sign.”

10. A Helpful Formula Table

Function	Sum	Difference
Sine	sin A × cos B + cos A × sin B	sin A × cos B − cos A × sin B
Cosine	cos A × cos B − sin A × sin B	cos A × cos B + sin A × sin B
Tangent	(tan A + tan B) ÷ (1 − tan A × tan B)	(tan A − tan B) ÷ (1 + tan A × tan B)

Mobile table tip: On a small phone screen, swipe horizontally inside the table to see the complete formulas.

11. Common Mistakes to Avoid

1. Using the wrong identity. Check whether the expression contains a sum or difference.
2. Forgetting the cosine sign change. Cosine is the identity most likely to cause sign errors.
3. Mixing up the order. sin(A + B) is not the same formula as cos(A + B).
4. Using decimal approximations too early. Keep exact values such as √2 ÷ 2 until the final step whenever possible.
5. Using the wrong angle decomposition. Choose familiar angles such as 30°, 45°, and 60° whenever they work.
6. Forgetting calculator mode. If checking a degree answer numerically, make sure your calculator is in degree mode.
7. Skipping a sign check. For example, cos(105°) should be negative because 105° lies in Quadrant II.

12. How to Choose the Best Angle Decomposition

When you see an unusual angle, look for familiar angles that add or subtract to it.

Angle	Useful Breakdown
15°	45° − 30°
75°	45° + 30°
105°	60° + 45°
120°	60° + 60°
165°	120° + 45°

Mobile tip: Swipe horizontally inside the table on a narrow phone screen if needed.

13. Practice Questions

Try these problems on your own before checking the answer section.

1. Find the exact value of sin(15°).
2. Find the exact value of cos(75°).
3. Find the exact value of sin(105°).
4. Find the exact value of cos(120°) using a sum identity.
5. Find the exact value of tan(15°).
6. Find the exact value of tan(105°).

14. Practice Answers

1. sin(15°): (√6 − √2) ÷ 4
2. cos(75°): (√6 − √2) ÷ 4
3. sin(105°): (√6 + √2) ÷ 4
4. cos(120°): −1 ÷ 2
5. tan(15°): 2 − √3
6. tan(105°): −(2 + √3)

15. Quick Review Checklist

Before solving:

• Can I rewrite the angle using familiar angles?
• Am I using sine, cosine, or tangent?
• Is the angle a sum or a difference?
• Did I copy the signs correctly?
• Can I keep the answer exact instead of rounding early?
• Does the final sign make sense based on the angle's quadrant?

16. Final Takeaway

The winning strategy is:

Break the angle → Choose the identity → Substitute → Simplify → Check

Sum and difference identities may seem like a lot of formulas at first, but they become much easier once you recognize their patterns. Focus especially on the sign change in the cosine identities and practice breaking unusual angles into familiar ones.

Remember: Sine keeps the sign pattern, while cosine switches it. With that memory tip and regular practice, complicated-looking trigonometric expressions become much more manageable.

Master the identities, and the angles become easier!

Practice exact values first, then use a calculator to check your work.

This tutorial is suitable for high school trigonometry, Algebra 2, precalculus, homeschool mathematics, and introductory college mathematics.