Sum and Difference Identities - Formulas & Applications

What Are Sum & Difference Identities?

The foundation for advanced trigonometry

The sum and difference identities allow us to express the sine, cosine, or tangent of a sum or difference of two angles in terms of the trig functions of the individual angles.

The Core Idea:

\sin(A + B) \neq \sin A + \sin B

Instead, we need special formulas to expand compound angles!

These identities are foundational for deriving double-angle formulas, half-angle formulas, product-to-sum formulas, and solving many trigonometric equations.

Learning Objectives

State and apply the sum and difference formulas for sine.
State and apply the sum and difference formulas for cosine.
State and apply the sum and difference formulas for tangent.
Use these identities to find exact values of non-standard angles.
Simplify trigonometric expressions using sum/difference formulas.
Prove other identities using these fundamental formulas.
Apply these identities to solve real-world problems.

Key Terminology

Sum Identity

Formula expressing trig function of (A + B) using trig functions of A and B.

Difference Identity

Formula expressing trig function of (A - B) using trig functions of A and B.

Compound Angle

An angle expressed as a sum or difference of other angles.

Exact Value

Value expressed using radicals and fractions, not decimal approximations.

Cofunction

sin(90° - θ) = cos(θ) relationship, derivable from difference formulas.

Standard Angles Reference

Memorize these values to use sum/difference formulas effectively:

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3
90°	1	0	undefined

Essential Formulas

Sine Sum

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

Same sign as the angle sum.

Sine Difference

\sin(A - B) = \sin A \cos B - \cos A \sin B

Same sign as the angle difference.

Cosine Sum

\cos(A + B) = \cos A \cos B - \sin A \sin B

Sign CHANGES: plus becomes minus.

Cosine Difference

\cos(A - B) = \cos A \cos B + \sin A \sin B

Sign CHANGES: minus becomes plus.

Tangent Sum

\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}

Valid when denominator ≠ 0.

Tangent Difference

\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}

Valid when denominator ≠ 0.

Memory Tricks

Sine Formula Pattern

"Sine keeps the sign" - The ± in sin(A ± B) matches the middle sign.

sin(A + B) → + in middle

sin(A - B) → - in middle

Cosine Formula Pattern

"Cosine changes the sign" - The ± in cos(A ± B) becomes ∓.

cos(A + B) → - in middle

cos(A - B) → + in middle

Term Pattern

• Sine: sin·cos + cos·sin (alternating functions)
• Cosine: cos·cos - sin·sin (same functions together)
• Tangent: fraction with sum/difference on top, product term on bottom

Worked Examples

Example 1

Finding sin(75°)

Problem:

Find the exact value of sin(75°).

Solution:

Step 1: Express 75° as a sum of standard angles:

75° = 45° + 30°

Step 2: Apply sin(A + B) formula:

\sin(75°) = \sin(45° + 30°)

= \sin 45° \cos 30° + \cos 45° \sin 30°

Step 3: Substitute known values:

= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}

= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}

Example 2

Finding cos(15°)

Problem:

Find the exact value of cos(15°).

Solution:

Step 1: Express 15° as a difference:

15° = 45° - 30°

Step 2: Apply cos(A - B) formula (sign changes to +):

\cos(15°) = \cos(45° - 30°)

= \cos 45° \cos 30° + \sin 45° \sin 30°

Step 3: Substitute and simplify:

= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}

Notice: sin(75°) = cos(15°) because they are complementary!

Example 3

Finding tan(105°)

Problem:

Find the exact value of tan(105°).

Solution:

Step 1: Express 105° as 60° + 45°:

\tan(105°) = \tan(60° + 45°)

Step 2: Apply tan(A + B) formula:

= \frac{\tan 60° + \tan 45°}{1 - \tan 60° \tan 45°}

= \frac{\sqrt{3} + 1}{1 - \sqrt{3} \cdot 1} = \frac{\sqrt{3} + 1}{1 - \sqrt{3}}

Step 3: Rationalize:

= \frac{(\sqrt{3} + 1)(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} = \frac{3 + 2\sqrt{3} + 1}{1 - 3} = \frac{4 + 2\sqrt{3}}{-2} = -2 - \sqrt{3}

Example 4

Simplifying an Expression

Problem:

Simplify: sin(x + 30°) cos(30°) - cos(x + 30°) sin(30°)

Solution:

Recognize the pattern: sin A cos B - cos A sin B = sin(A - B)

Let A = x + 30° and B = 30°:

\sin(x + 30°)\cos 30° - \cos(x + 30°)\sin 30°

= \sin[(x + 30°) - 30°] = \sin(x)

Example 5

Proving an Identity

Problem:

Prove: cos(90° - θ) = sin(θ)

Solution:

Use cos(A - B) = cos A cos B + sin A sin B:

\cos(90° - \theta) = \cos 90° \cos \theta + \sin 90° \sin \theta

= 0 \cdot \cos \theta + 1 \cdot \sin \theta = \sin \theta

This is the cofunction identity! ✓

Example 6

Using Given Information

Problem:

If sin A = 3/5 (A in Q1) and cos B = 5/13 (B in Q1), find sin(A + B).

Solution:

Step 1: Find missing values using Pythagorean identity:

\cos A = \sqrt{1 - \sin^2 A} = \sqrt{1 - 9/25} = \frac{4}{5}

\sin B = \sqrt{1 - \cos^2 B} = \sqrt{1 - 25/169} = \frac{12}{13}

Step 2: Apply sin(A + B):

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

= \frac{3}{5} \cdot \frac{5}{13} + \frac{4}{5} \cdot \frac{12}{13}

= \frac{15}{65} + \frac{48}{65} = \frac{63}{65}

More Worked Examples

Example 7

Simplifying Expression

Problem:

Simplify: cos(x + y)cos(y) + sin(x + y)sin(y)

Solution:

\cos(x+y)\cos y + \sin(x+y)\sin y

= \cos[(x+y) - y] \quad \text{(reverse cos difference)}

= \cos x

Example 8

Sum of Three Angles

Problem:

Find tan(A + B + C) if tan A = 1, tan B = 2, tan C = 3.

Solution:

\tan(A+B+C) = \frac{\tan A + \tan B + \tan C - \tan A \tan B \tan C}{1 - \tan A \tan B - \tan B \tan C - \tan A \tan C}

= \frac{1 + 2 + 3 - (1)(2)(3)}{1 - 2 - 6 - 3} = \frac{6 - 6}{-10} = 0

Example 9

Phase Shift Application

Problem:

Express 3sin x + 4cos x in the form R sin(x + φ).

Solution:

Step 1: Find R:

R = \sqrt{3^2 + 4^2} = 5

Step 2: Find φ:

\tan\phi = \frac{4}{3}, \quad \phi = \arctan\frac{4}{3} \approx 53.13°

Answer: 3sin x + 4cos x = 5 sin(x + 53.13°)

Example 10

Proving Identity

Problem:

Prove: sin(A + B)sin(A - B) = sin²A - sin²B

Solution:

\sin(A+B)\sin(A-B)

= (\sin A\cos B + \cos A\sin B)(\sin A\cos B - \cos A\sin B)

= \sin^2 A\cos^2 B - \cos^2 A\sin^2 B

= \sin^2 A(1-\sin^2 B) - (1-\sin^2 A)\sin^2 B

= \sin^2 A - \sin^2 B \quad \checkmark

Understanding the Derivations

Cosine Difference Formula Proof (Geometric)

Consider two points P and Q on the unit circle at angles A and B from positive x-axis.
P = (cos A, sin A) and Q = (cos B, sin B)
Distance PQ can be calculated using distance formula.
Also, PQ² = 2 - 2cos(A - B) using Law of Cosines on triangle OPQ.
Equating both expressions and simplifying gives the formula.

Deriving Other Formulas

• cos(A + B): Use cos(A - (-B)) = cos(A + B)
• sin formulas: Use sin θ = cos(90° - θ) cofunction identity
• tan formulas: Use tan = sin/cos and divide the formulas

Complete Formula Reference

Sine Formulas

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

\sin(A - B) = \sin A\cos B - \cos A\sin B

Cosine Formulas

\cos(A + B) = \cos A\cos B - \sin A\sin B

\cos(A - B) = \cos A\cos B + \sin A\sin B

Tangent Formulas

\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A\tan B}

\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A\tan B}

Common Mistakes to Avoid

Wrong sign in cosine formula

cos(A + B) uses MINUS in the middle. cos(A - B) uses PLUS. The sign changes!

Mixing up sine and cosine terms

sin uses 'sin cos + cos sin' pattern. cos uses 'cos cos - sin sin' pattern.

Forgetting tangent restrictions

tan(A ± B) is undefined when tanA tanB = ±1 (denominator = 0).

Not simplifying completely

Always check if your answer can be simplified using known values or other identities.

Wrong angle decomposition

75° = 45° + 30° (not 60° + 15°, though that also works). Choose convenient angles.

Real-World Applications

📡

Signal Processing

Combining waves of different frequencies.

🔬

Physics

Analyzing interference patterns in light and sound.

⚡

Engineering

Phase angle calculations in AC circuits.

🎵

Music

Understanding harmonics and beat frequencies.

🧭

Navigation

Calculating bearings and course corrections.

🎮

Computer Graphics

Rotation transformations in 2D and 3D.

Quick Reference Card

Sine

sin(A+B) = sinA cosB + cosA sinB

sin(A-B) = sinA cosB - cosA sinB

Cosine

cos(A+B) = cosA cosB - sinA sinB

cos(A-B) = cosA cosB + sinA sinB

Tangent

tan(A+B) = (tanA+tanB)/(1-tanA tanB)

tan(A-B) = (tanA-tanB)/(1+tanA tanB)

Frequently Asked Questions

Why are these called 'sum and difference' identities?

They express trig functions of (A + B) or (A - B) in terms of trig functions of A and B separately.

How do I remember the sine formula signs?

sin(A ± B) = sinA cosB ± cosA sinB. The sign in the middle matches the ± in the angle. Think: 'Sine keeps the sign.'

How do I remember the cosine formula signs?

cos(A ± B) = cosA cosB ∓ sinA sinB. The sign CHANGES! Plus becomes minus, minus becomes plus. Think: 'Cosine changes the sign.'

When should I use these formulas?

Use them to: (1) Find exact values like sin(75°), (2) Simplify expressions, (3) Prove other identities, (4) Solve equations.

What angles can I break down?

Express angles as sums/differences of standard angles: 30°, 45°, 60°, 90°, etc. For example: 75° = 45° + 30°, 15° = 45° - 30°.

Why is the tangent formula more complex?

tan(A ± B) has a denominator 1 ∓ tanA tanB that can become undefined when the product tanA tanB equals ±1.

How do these relate to double angle formulas?

Double angle formulas are special cases! sin(2A) = sin(A + A), cos(2A) = cos(A + A).

Practice Problems Preview

Test your understanding:

Find the exact value of sin(105°).
Find the exact value of cos(75°).
Find tan(15°) using difference formula.
Simplify: cos(x) cos(2x) + sin(x) sin(2x)
If sin A = 4/5 (Q1) and sin B = 5/13 (Q2), find cos(A + B).
Prove: sin(π/2 + x) = cos(x)
Prove: tan(A + B + C) formula for three angles.
Find sin(A - B) if tan A = 3/4 and tan B = 1/2.
Simplify sin(x + 60°) + sin(x - 60°).

Additional Worked Examples

Special Angles

Find sin(195°)

Solution:

\sin 195° = \sin(180° + 15°) = -\sin 15°

= -\sin(45° - 30°) = -(\sin 45°\cos 30° - \cos 45°\sin 30°)

= -\left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}\right)

= -\frac{\sqrt{6} - \sqrt{2}}{4}

Cofunction

Prove cos(π/2 - x) = sin(x)

Proof:

\cos\left(\frac{\pi}{2} - x\right) = \cos\frac{\pi}{2}\cos x + \sin\frac{\pi}{2}\sin x

= 0 \cdot \cos x + 1 \cdot \sin x = \sin x \quad \checkmark

Formula Summary Table

Function	Sum Formula	Difference Formula
Sine	sin α cos β + cos α sin β	sin α cos β - cos α sin β
Cosine	cos α cos β - sin α sin β	cos α cos β + sin α sin β
Tangent	(tan α + tan β)/(1 - tan α tan β)	(tan α - tan β)/(1 + tan α tan β)

Practice & Tools

Trig Calculator

Verify your calculations

Practice Questions

Test your identity skills

Next Steps

Double Angle Formulas

Special case of sum formulas

Identity Transformations

Advanced simplifications

Triangle Solving

Apply to real problems