Mastory | Notable Identities: Complete Guide with Formulas and Examples

Notable identities can be considered important tools for mathematical reasoning. These are equations that hold true for every value within the expression's domain of definition, meaning the identity is true for every number where the expression is meaningful at all. So these are eternal truths that can also be viewed as demonstrations of purely formal differences, like the identity between ½ and 2/4: ½ = 2/4 even though the numbers on both sides of the equation don't look exactly the same. Identities help us transform algebraic expressions that arise during problem solving into the most suitable form for the solution at hand.

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Key Take-aways About Notable Mathematical Identities:

Identity vs. Equation: Identities are always true for every value within the domain of definition (e.g., (a+b)² = a² + 2ab + b²), while equations are not necessarily so. Identities can be discovered or established as existing mathematical truths, while we write equations in order to solve them—that is, to find those values for which they are true.
Algebraic Foundations: Among the most important formulas are the square of sum/difference and the product identity (a+b)(a-b) = a² - b²
Trigonometric Basics: sin²α + cos²α = 1 and the addition formulas (sin(α+β), cos(α+β)) are indispensable for all trigonometric calculations
Logarithm Rules: log(a·b) = log(a) + log(b) and log(a^n) = n·log(a) - without these, logarithmic equations cannot be solved
Practical Application: Notable identities are used not only in math class but also in architecture, physics, and financial calculations
Learning Tip: Understanding the derivation is more important than memorization - this way formulas stick long-term, and if you forget them, you can re-derive them yourself.
Common Mistake: (a+b)² ≠ a² + b² - always remember the 2ab middle term!

What is an Identity in Mathematics?

An identity is an equality that holds true for all values of the variables (within the expression's domain of definition). This fundamentally differs from equations that are only satisfied by certain values. For example, x² - 1 = (x-1)(x+1) is an identity because it's true for any value of x, while the equation x² = 4 is only satisfied by x = 2 and x = -2.

The "notable" designation indicates that these identities are particularly important and frequently used in solving mathematical problems. Knowledge of them helps in simplifying expressions, solving equations, and discovering deeper relationships.

We denote identities as follows:

General form: A ≡ B

But we can also simply use the equals sign: A = B

The difference between an equation and an identity is therefore: while an equation is a "question" (for which x values does it hold?), an identity is a "statement" (this is always true). However, identities are also equations, since we can certainly ask the question: for which x values does the equation (x+2)(x-2)=x²-4 hold? The answer will naturally be: for all of them, and from this we can see that we've stumbled upon an identity.

Algebraic Notable Identities

Binomial Theorem and Squaring

Among the most well-known notable identities are the squares of binomial expressions:

Square of a sum: (a+b)² = a² + 2ab + b²

Square of a difference: (a-b)² = a² - 2ab + b²

These formulas are extremely useful for simplifying or expanding algebraic expressions. For example, expanding (x+3)² using the notable identity: x² + 2·x·3 + 3² = x² + 6x + 9.

We can see that these identities simply save us one intermediate step, since if we simply calculate the square of the expression (a+b) using the definition of squaring, we can see that we reach the right side with just one intermediate step:

(a+b)² = (a+b)(a+b) = aa+ab+ba+bb = a²+2ab+b²

Their real usefulness therefore mainly shows in the other direction: when we're dealing with a fractional expression like: (a²+b²-2ab)/(a-b)

In this case, the identity really helps a lot, since we can calculate this expression with their help even if we've never heard of the rules for polynomial division.

We can also visualize the square of a sum with a geometric diagram, where an a×a square and a b×b square, along with two a×b rectangles together form the (a+b)×(a+b) square.

Product-Sum Identities

One of the most fundamental and frequently used notable identities is the "product of difference and sum":

(a+b)(a-b) = a² - b²

This formula is an excellent tool for simplifying expressions leading to difference of squares. For example, (x+5)(x-5) = x² - 25.

Other important identities are the cubes of binomial expressions:

(a+b)³ = a³ + 3a²b + 3ab² + b³

(a-b)³ = a³ - 3a²b + 3ab² - b³

These identities are special cases of the binomial theorem, which provides a general formula for calculating the nth power: (a+b)^n = Σ(k=0→n) (n!/(k!(n-k)!)) · a^(n-k) · b^k

Power Identities

The fundamental identities for powers are:

a^m · a^n = a^(m+n)
(a^m)^n = a^(m·n)
(a·b)^n = a^n · b^n
a^m / a^n = a^(m-n), where a ≠ 0

For example, calculating x³ · x²: x³ · x² = x^(3+2) = x^5

These identities are easily understood using the definition, namely that exponentiation means multiplication by itself. Since the logarithm is precisely the search for the exponent for a given power base, these identities are also fundamental tools for solving logarithmic equations. Additionally, the laws of exponentiation are closely related to the properties of roots and logarithms.

Trigonometric Notable Identities

Pythagorean Theorem and Basic Relationships

The foundation of trigonometry is provided by the properties of right triangles and the unit circle. One of the most fundamental notable identities in trigonometry is:

sin²α + cos²α = 1

This relationship can be derived directly from the Pythagorean theorem when we consider sine and cosine values on the unit circle.

The definitions of tangent and cotangent and related identities:

tanα = sinα / cosα
cotα = cosα / sinα
tan²α + 1 = 1/cos²α (or sec²α)
cot²α + 1 = 1/sin²α (or cosec²α)

Addition Formulas

Another important group in trigonometry consists of the addition formulas:

sin(α+β) = sinα·cosβ + cosα·sinβ
sin(α-β) = sinα·cosβ - cosα·sinβ
cos(α+β) = cosα·cosβ - sinα·sinβ
cos(α-β) = cosα·cosβ + sinα·sinβ
tan(α+β) = (tanα + tanβ) / (1 - tanα·tanβ)
tan(α-β) = (tanα - tanβ) / (1 + tanα·tanβ)

These formulas enable the calculation of trigonometric function values for compound angles. For example, if we know the values of sin(30°) and sin(45°), we can easily calculate the value of sin(75°) using the addition formula.

Double and Half-Angle Formulas

Double angle formulas:

sin(2α) = 2·sinα·cosα
cos(2α) = cos²α - sin²α = 2cos²α - 1 = 1 - 2sin²α
tan(2α) = 2tanα / (1 - tan²α)

Half-angle formulas:

sin(α/2) = ±√((1 - cosα) / 2)
cos(α/2) = ±√((1 + cosα) / 2)
tan(α/2) = (1 - cosα) / sinα = sinα / (1 + cosα)

These formulas are particularly useful in integration and solving trigonometric equations. Double angle formulas help simplify more complex trigonometric expressions, while half-angle formulas are often useful for calculating more complicated integrals.

Logarithm Identities

The most important identities for logarithmic operations:

log(a·b) = log(a) + log(b)
log(a/b) = log(a) - log(b)
log(a^n) = n·log(a)
log_a(b) = log_c(b) / log_c(a), for any base c

These identities follow from the definition of logarithms and are extremely useful for solving logarithmic equations and simplifying expressions.

For example, simplifying the expression log(2x·3y): log(2x·3y) = log(2x) + log(3y) = log(2) + log(x) + log(3) + log(y)

Additional useful logarithm identities:

log_a(1) = 0 (for any base)
log_a(a) = 1 (for any base)
log_a(b^c) = c·log_a(b)

Practical Applications and Examples

Notable identities are not merely theoretical curiosities but are extremely useful in practice. Let's look at some concrete examples:

Example 1: Calculate 99² mentally! Apply the formula (100-1)² = 100² - 2·100·1 + 1²: 99² = (100-1)² = 10000 - 200 + 1 = 9801

Example 2: Calculate the expression (2x+3)(2x-3)! Apply the formula (a+b)(a-b) = a² - b²: (2x+3)(2x-3) = (2x)² - 3² = 4x² - 9

Example 3: Express the value of sin(15°) using known trigonometric functions! We know that 15° = 45° - 30°, so we can use the difference formula: sin(15°) = sin(45° - 30°) = sin(45°)·cos(30°) - cos(45°)·sin(30°) = (√2/2)·(√3/2) - (√2/2)·(1/2) = √6/4 - √2/4 = (√6 - √2)/4

Example 4: Solve the equation x² - 6x + 9 = 0 using notable identities! We can recognize that x² - 6x + 9 = (x - 3)², so the equation can be written as (x - 3)² = 0, whose solution is x = 3.

Example 5: Calculate the value of log₂(24)-log₂(3) using logarithm identities!

log₂(24)-log₂(3) = log₂(24/3) = log₂(8) = log₂(2³) = 3·log₂(2) = 3·1 = 3

In real life, notable identities appear in many places:

In architecture and engineering work when applying the Pythagorean theorem and trigonometric identities
In physics when describing waves using trigonometric identities (e.g., describing harmonic oscillatory motion or wave interference)
In financial calculations using logarithmic and power identities (e.g., compound interest calculations)
In data analysis and statistics using applications of the binomial theorem (probability theory, sampling)
In computer graphics using trigonometric identities (rotations, reflections)

Effective Learning Methods for Notable Identities

To effectively master notable identities, we recommend the following methods:

Connection Points: Understand how individual identities relate to each other and how one can be derived from another. For example, the different forms of cos(2α) can all be derived from each other using the Pythagorean identity.

Derivation Practice: Try to derive notable identities from basic principles. This not only helps better understand the formulas but also develops mathematical thinking.

Teach Others: When you try to explain notable identities to someone, your own understanding deepens. Questions that arise during teaching often bring up new perspectives.

Active Recall: Instead of just reading the formulas, try to recall them from memory, then check your solution. Make cards with the expression on one side and the expanded form on the other.

Practical Application: Solve problems of varying difficulty that require the use of notable identities. Start with simple examples, then gradually progress to more complex problems.

Visual Representation: Create geometric diagrams to illustrate algebraic identities, or use graphs to understand trigonometric relationships. Visualization helps understand the meaning of formulas more deeply.

Regular Review: Apply spaced repetition method – return to previously learned identities at intervals. Optimal review times: after 1 day, after 3 days, after 1 week, after 2 weeks, then monthly.

Memorization Techniques: Use mnemonics, acronyms, or stories that help recall more complex formulas. For example, the form (a+b)² = a² + 2ab + b² is easier to remember if you notice the "square – double product – square" pattern.

Common Mistakes and Pitfalls

When applying notable identities, watch out for the following typical mistakes:

Confusing Squares: Many people incorrectly think that (a+b)² = a² + b². The correct formula is (a+b)² = a² + 2ab + b². This is one of the most common mistakes, which can easily be avoided by always checking the formula with a simple example (e.g., (2+3)² = 25, not 13).

Incorrect Sign Handling: With the square of a difference, a common mistake is getting the sign of the middle term wrong: (a-b)² = a² - 2ab + b², not a² + 2ab + b². It's important to remember that with the square of a difference, the middle term is negative.

Misunderstanding Powers: A common confusion is that (a+b)^n ≠ a^n + b^n, except when n=1. For example, (2+3)³ ≠ 2³ + 3³, but should be expanded according to the binomial theorem.

Mixing Trigonometric Formulas: Confusing sin(A+B) and sin(A)·sin(B) is a typical mistake. Remember that sin(A+B) = sin(A)·cos(B) + cos(A)·sin(B), while sin(A)·sin(B) is a simple product.

Incorrect Application of Logarithm Identities: It's important to remember that log(a+b) ≠ log(a) + log(b). The logarithm becomes a sum only for products: log(a·b) = log(a) + log(b).

Confusing Binomial Coefficients: When applying the binomial theorem, a common mistake is using incorrect coefficients. Pascal's triangle can help determine the appropriate coefficients.

Ignoring Domain of Definition: Certain identities are only valid under certain conditions. For example, with logarithmic identities, expressions must be positive, or with radical identities, attention must be paid to the non-negative condition.

Confusing Identities and Equations: Identities are true for all values (within the domain of definition), while equations are only true for certain values. This conceptual difference is important for correct application.

To avoid mistakes:

Create a summary note of all identities
Practice applying formulas with simple examples
Check your solutions in multiple ways (substitution, calculator)
If uncertain, derive an identity from the basics
Always check the identity with concrete numbers before applying it

Sources and Further Reading

Mastory: Mathematical Foundations
GeoGebra: Interactive Trigonometric Visualizations
Wolfram MathWorld: Trigonometric Identities
Paul's Online Math Notes: Logarithm Functions

Try the Mastory app and create personalized exercises in minutes, or explore our tutorials to discover all the features of this AI designed for math teachers!

Knowledge of notable identities not only helps you solve math problems but also develops your thinking skills and gives you tools useful across many scientific fields. Regular practice and understanding the connections between identities are key to mastering them.

Sources and Further Reading

Notable identities can be considered important tools for mathematical reasoning. These are equations that hold true for every value within the expression's domain of definition, meaning the identity is true for every number where the expression is meaningful at all. So these are eternal truths that can also be viewed as demonstrations of purely formal differences, like the identity between ½ and 2/4: ½ = 2/4 even though the numbers on both sides of the equation don't look exactly the same. Identities help us transform algebraic expressions that arise during problem solving into the most suitable form for the solution at hand.

If you're not looking to study but want awesome practice problems, register for free on Mastory and create custom exercises in moments, or browse our tutorials to discover the full range of AI tools designed for math teachers!

Key Take-aways About Notable Mathematical Identities:

Identity vs. Equation: Identities are always true for every value within the domain of definition (e.g., (a+b)² = a² + 2ab + b²), while equations are not necessarily so. Identities can be discovered or established as existing mathematical truths, while we write equations in order to solve them—that is, to find those values for which they are true.
Algebraic Foundations: Among the most important formulas are the square of sum/difference and the product identity (a+b)(a-b) = a² - b²
Trigonometric Basics: sin²α + cos²α = 1 and the addition formulas (sin(α+β), cos(α+β)) are indispensable for all trigonometric calculations
Logarithm Rules: log(a·b) = log(a) + log(b) and log(a^n) = n·log(a) - without these, logarithmic equations cannot be solved
Practical Application: Notable identities are used not only in math class but also in architecture, physics, and financial calculations
Learning Tip: Understanding the derivation is more important than memorization - this way formulas stick long-term, and if you forget them, you can re-derive them yourself.
Common Mistake: (a+b)² ≠ a² + b² - always remember the 2ab middle term!

What is an Identity in Mathematics?

An identity is an equality that holds true for all values of the variables (within the expression's domain of definition). This fundamentally differs from equations that are only satisfied by certain values. For example, x² - 1 = (x-1)(x+1) is an identity because it's true for any value of x, while the equation x² = 4 is only satisfied by x = 2 and x = -2.

The "notable" designation indicates that these identities are particularly important and frequently used in solving mathematical problems. Knowledge of them helps in simplifying expressions, solving equations, and discovering deeper relationships.

We denote identities as follows:

General form: A ≡ B

But we can also simply use the equals sign: A = B

The difference between an equation and an identity is therefore: while an equation is a "question" (for which x values does it hold?), an identity is a "statement" (this is always true). However, identities are also equations, since we can certainly ask the question: for which x values does the equation (x+2)(x-2)=x²-4 hold? The answer will naturally be: for all of them, and from this we can see that we've stumbled upon an identity.

Algebraic Notable Identities

Binomial Theorem and Squaring

Among the most well-known notable identities are the squares of binomial expressions:

Square of a sum: (a+b)² = a² + 2ab + b²

Square of a difference: (a-b)² = a² - 2ab + b²

These formulas are extremely useful for simplifying or expanding algebraic expressions. For example, expanding (x+3)² using the notable identity: x² + 2·x·3 + 3² = x² + 6x + 9.

We can see that these identities simply save us one intermediate step, since if we simply calculate the square of the expression (a+b) using the definition of squaring, we can see that we reach the right side with just one intermediate step:

(a+b)² = (a+b)(a+b) = aa+ab+ba+bb = a²+2ab+b²

Their real usefulness therefore mainly shows in the other direction: when we're dealing with a fractional expression like: (a²+b²-2ab)/(a-b)

In this case, the identity really helps a lot, since we can calculate this expression with their help even if we've never heard of the rules for polynomial division.

We can also visualize the square of a sum with a geometric diagram, where an a×a square and a b×b square, along with two a×b rectangles together form the (a+b)×(a+b) square.

Product-Sum Identities

One of the most fundamental and frequently used notable identities is the "product of difference and sum":

(a+b)(a-b) = a² - b²

This formula is an excellent tool for simplifying expressions leading to difference of squares. For example, (x+5)(x-5) = x² - 25.

Other important identities are the cubes of binomial expressions:

(a+b)³ = a³ + 3a²b + 3ab² + b³

(a-b)³ = a³ - 3a²b + 3ab² - b³

These identities are special cases of the binomial theorem, which provides a general formula for calculating the nth power: (a+b)^n = Σ(k=0→n) (n!/(k!(n-k)!)) · a^(n-k) · b^k

Power Identities

The fundamental identities for powers are:

a^m · a^n = a^(m+n)
(a^m)^n = a^(m·n)
(a·b)^n = a^n · b^n
a^m / a^n = a^(m-n), where a ≠ 0

For example, calculating x³ · x²: x³ · x² = x^(3+2) = x^5

These identities are easily understood using the definition, namely that exponentiation means multiplication by itself. Since the logarithm is precisely the search for the exponent for a given power base, these identities are also fundamental tools for solving logarithmic equations. Additionally, the laws of exponentiation are closely related to the properties of roots and logarithms.

Trigonometric Notable Identities

Pythagorean Theorem and Basic Relationships

The foundation of trigonometry is provided by the properties of right triangles and the unit circle. One of the most fundamental notable identities in trigonometry is:

sin²α + cos²α = 1

This relationship can be derived directly from the Pythagorean theorem when we consider sine and cosine values on the unit circle.

The definitions of tangent and cotangent and related identities:

tanα = sinα / cosα
cotα = cosα / sinα
tan²α + 1 = 1/cos²α (or sec²α)
cot²α + 1 = 1/sin²α (or cosec²α)

Addition Formulas

Another important group in trigonometry consists of the addition formulas:

sin(α+β) = sinα·cosβ + cosα·sinβ
sin(α-β) = sinα·cosβ - cosα·sinβ
cos(α+β) = cosα·cosβ - sinα·sinβ
cos(α-β) = cosα·cosβ + sinα·sinβ
tan(α+β) = (tanα + tanβ) / (1 - tanα·tanβ)
tan(α-β) = (tanα - tanβ) / (1 + tanα·tanβ)

These formulas enable the calculation of trigonometric function values for compound angles. For example, if we know the values of sin(30°) and sin(45°), we can easily calculate the value of sin(75°) using the addition formula.

Double and Half-Angle Formulas

Double angle formulas:

sin(2α) = 2·sinα·cosα
cos(2α) = cos²α - sin²α = 2cos²α - 1 = 1 - 2sin²α
tan(2α) = 2tanα / (1 - tan²α)

Half-angle formulas:

sin(α/2) = ±√((1 - cosα) / 2)
cos(α/2) = ±√((1 + cosα) / 2)
tan(α/2) = (1 - cosα) / sinα = sinα / (1 + cosα)

These formulas are particularly useful in integration and solving trigonometric equations. Double angle formulas help simplify more complex trigonometric expressions, while half-angle formulas are often useful for calculating more complicated integrals.

Logarithm Identities

The most important identities for logarithmic operations:

log(a·b) = log(a) + log(b)
log(a/b) = log(a) - log(b)
log(a^n) = n·log(a)
log_a(b) = log_c(b) / log_c(a), for any base c

These identities follow from the definition of logarithms and are extremely useful for solving logarithmic equations and simplifying expressions.

For example, simplifying the expression log(2x·3y): log(2x·3y) = log(2x) + log(3y) = log(2) + log(x) + log(3) + log(y)

Additional useful logarithm identities:

log_a(1) = 0 (for any base)
log_a(a) = 1 (for any base)
log_a(b^c) = c·log_a(b)

Practical Applications and Examples

Notable identities are not merely theoretical curiosities but are extremely useful in practice. Let's look at some concrete examples:

Example 1: Calculate 99² mentally! Apply the formula (100-1)² = 100² - 2·100·1 + 1²: 99² = (100-1)² = 10000 - 200 + 1 = 9801

Example 2: Calculate the expression (2x+3)(2x-3)! Apply the formula (a+b)(a-b) = a² - b²: (2x+3)(2x-3) = (2x)² - 3² = 4x² - 9

Example 3: Express the value of sin(15°) using known trigonometric functions! We know that 15° = 45° - 30°, so we can use the difference formula: sin(15°) = sin(45° - 30°) = sin(45°)·cos(30°) - cos(45°)·sin(30°) = (√2/2)·(√3/2) - (√2/2)·(1/2) = √6/4 - √2/4 = (√6 - √2)/4

Example 4: Solve the equation x² - 6x + 9 = 0 using notable identities! We can recognize that x² - 6x + 9 = (x - 3)², so the equation can be written as (x - 3)² = 0, whose solution is x = 3.

Example 5: Calculate the value of log₂(24)-log₂(3) using logarithm identities!

log₂(24)-log₂(3) = log₂(24/3) = log₂(8) = log₂(2³) = 3·log₂(2) = 3·1 = 3

In real life, notable identities appear in many places:

In architecture and engineering work when applying the Pythagorean theorem and trigonometric identities
In physics when describing waves using trigonometric identities (e.g., describing harmonic oscillatory motion or wave interference)
In financial calculations using logarithmic and power identities (e.g., compound interest calculations)
In data analysis and statistics using applications of the binomial theorem (probability theory, sampling)
In computer graphics using trigonometric identities (rotations, reflections)