1. Fundamental Identities

Fundamental Trigonometric Identities

Trigonometric identities are mathematical equations that describe relationships between trigonometric functions. Unlike ordinary equations, which may only be true for particular values of a variable, trigonometric identities are true for every angle where both sides are defined.

These identities allow us to rewrite complicated expressions, solve trigonometric equations, verify mathematical relationships, and simplify calculations.

The three main groups of fundamental trigonometric identities are:

• Reciprocal identities: Relationships between trigonometric functions and their reciprocals.

• Quotient identities: Relationships expressing tangent and cotangent in terms of sine and cosine.

• Pythagorean identities: Relationships derived from the Pythagorean theorem and the unit circle.

Understanding these identities is essential for advanced trigonometry, calculus, physics, and engineering.

What Is a Trigonometric Identity?

A trigonometric identity is an equation involving trigonometric functions that is true for all values of the variable for which both sides are defined.

For example:

sin² θ + cos² θ = 1

This equation is true for every real angle θ.

At θ = 30°:

sin² 30° + cos² 30°

= (1/2)² + (√3/2)²

= 1/4 + 3/4

= 1

At θ = 45°:

sin² 45° + cos² 45°

= (√2/2)² + (√2/2)²

= 1/2 + 1/2

= 1

The relationship continues to hold for all angles.

Identities Compared with Equations

Consider:

sin θ = 1/2

This is a trigonometric equation. It is true only for certain angles.

However:

sin² θ + cos² θ = 1

is an identity because it is always true.

Identities are useful because they allow one expression to be replaced by another equivalent expression.

Reciprocal Identities

The reciprocal identities connect the six trigonometric functions.

The three fundamental reciprocal relationships are:

csc θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

These can also be written as:

sin θ = 1/csc θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Each identity is valid wherever both sides are defined.

Reciprocal Identity Table

Worked Example: Using a Reciprocal Identity

Suppose:

sin θ = 3/5

Find csc θ.

Using:

csc θ = 1/sin θ

Substitute:

csc θ = 1/(3/5)

Therefore:

csc θ = 5/3

Worked Example: Finding Secant

Suppose:

cos θ = −4/7

Find sec θ.

sec θ = 1/cos θ

sec θ = 1/(−4/7)

sec θ = −7/4

The reciprocal has the same sign as the original nonzero value.

Understanding Reciprocal Relationships

Reciprocal identities follow directly from right-triangle trigonometry.

For an acute angle θ:

sin θ = opposite/hypotenuse

Taking the reciprocal gives:

1/sin θ = hypotenuse/opposite

This is the definition of cosecant.

Therefore:

csc θ = 1/sin θ

Similarly:

cos θ = adjacent/hypotenuse

Taking the reciprocal:

sec θ = hypotenuse/adjacent

And:

tan θ = opposite/adjacent

Taking the reciprocal:

cot θ = adjacent/opposite

Important Restrictions

Reciprocal identities involve division.

Division by zero is undefined.

Therefore:

• Cosecant is undefined when sine equals zero.

• Secant is undefined when cosine equals zero.

• Cotangent is undefined when sine equals zero.

For example:

csc 0° = 1/sin 0°

= 1/0

This is undefined.

Likewise:

sec 90° = 1/cos 90°

= 1/0

This is undefined.

The expression 1/tan θ also requires tangent to be defined and nonzero.

Quotient Identities

Quotient identities express tangent and cotangent using sine and cosine.

The two fundamental quotient identities are:

tan θ = sin θ/cos θ

cot θ = cos θ/sin θ

These identities are particularly useful because they allow expressions involving tangent and cotangent to be rewritten using only sine and cosine.

Deriving the Tangent Identity

From right-triangle trigonometry:

sin θ = opposite/hypotenuse

cos θ = adjacent/hypotenuse

Divide sine by cosine:

sin θ/cos θ

= (opposite/hypotenuse)/(adjacent/hypotenuse)

The hypotenuse factors cancel.

This leaves:

opposite/adjacent

which equals tangent.

Therefore:

tan θ = sin θ/cos θ

Deriving the Cotangent Identity

Similarly:

cos θ/sin θ

= (adjacent/hypotenuse)/(opposite/hypotenuse)

= adjacent/opposite

Therefore:

cot θ = cos θ/sin θ

Applying Quotient Identities

Worked Example: Finding Tangent

Suppose:

sin θ = 5/13

cos θ = 12/13

Find tan θ.

Use:

tan θ = sin θ/cos θ

Substitute:

tan θ = (5/13)/(12/13)

Simplify:

tan θ = 5/12

Worked Example: Finding Cotangent

Suppose:

sin θ = −3/5

cos θ = 4/5

Find cot θ.

cot θ = cos θ/sin θ

= (4/5)/(−3/5)

cot θ = −4/3

The negative result is consistent with an angle in Quadrant IV.

Simplifying an Expression

Simplify:

sin θ × cot θ

Replace cotangent using the quotient identity:

sin θ × (cos θ/sin θ)

Cancel sin θ:

= cos θ

This simplification is valid where sin θ ≠ 0.

The Fundamental Pythagorean Identity

The most important trigonometric identity is:

sin² θ + cos² θ = 1

This relationship comes directly from the unit circle.

The equation of the unit circle is:

x² + y² = 1

For a point on the unit circle:

x = cos θ

y = sin θ

Substituting gives:

cos² θ + sin² θ = 1

Therefore:

sin² θ + cos² θ = 1

Connection to the Pythagorean Theorem

For a right triangle:

a² + b² = c²

On the unit circle, the hypotenuse has length 1.

The horizontal and vertical components are cos θ and sin θ.

Therefore:

(cos θ)² + (sin θ)² = 1²

This produces the fundamental Pythagorean identity.

Rearranging the Pythagorean Identity

Starting with:

sin² θ + cos² θ = 1

We can rearrange to obtain:

sin² θ = 1 − cos² θ

and:

cos² θ = 1 − sin² θ

These forms are useful when one trigonometric value is known and another must be calculated.

Worked Example

Suppose:

sin θ = 3/5

and θ lies in Quadrant I.

Find cos θ.

Use:

sin² θ + cos² θ = 1

Substitute:

(3/5)² + cos² θ = 1

9/25 + cos² θ = 1

cos² θ = 16/25

Take the square root:

cos θ = ±4/5

Since θ lies in Quadrant I, cosine is positive.

Therefore:

cos θ = 4/5

Why the Quadrant Matters

When taking the square root of an equation, both positive and negative values may be possible.

The quadrant determines the correct sign.

For example, if the same sine value occurred in Quadrant II:

cos θ = −4/5

The Second Pythagorean Identity

Starting with:

sin² θ + cos² θ = 1

Divide every term by cos² θ:

sin² θ/cos² θ + cos² θ/cos² θ = 1/cos² θ

Using quotient and reciprocal identities:

sin² θ/cos² θ = tan² θ

cos² θ/cos² θ = 1

1/cos² θ = sec² θ

Therefore:

1 + tan² θ = sec² θ

This is the second Pythagorean identity.

It is valid where cos θ ≠ 0.

Rearranged Forms

tan² θ = sec² θ − 1

1 = sec² θ − tan² θ

Worked Example

Suppose:

tan θ = 2

Find sec² θ.

Use:

1 + tan² θ = sec² θ

Substitute:

1 + 2² = sec² θ

1 + 4 = sec² θ

sec² θ = 5

The Third Pythagorean Identity

Again, begin with:

sin² θ + cos² θ = 1

Divide every term by sin² θ:

sin² θ/sin² θ + cos² θ/sin² θ = 1/sin² θ

Simplify:

1 + cot² θ = csc² θ

Therefore:

1 + cot² θ = csc² θ

This is the third Pythagorean identity.

It is valid where sin θ ≠ 0.

Rearranged Forms

cot² θ = csc² θ − 1

1 = csc² θ − cot² θ

Worked Example

Suppose:

cot θ = 3

Find csc² θ.

Use:

1 + cot² θ = csc² θ

Substitute:

1 + 3² = csc² θ

1 + 9 = csc² θ

csc² θ = 10

Summary of the Fundamental Identities

These identities form the foundation for most trigonometric simplification and verification problems.

Verifying Trigonometric Identities

To verify a trigonometric identity, we show that the expressions on both sides are equivalent.

Unlike solving an equation, we are not trying to find a particular value of θ.

Instead, we demonstrate that the relationship is true for every angle in its domain.

A common strategy is to begin with the more complicated side and simplify it until it matches the other side.

Worked Example: Verifying a Reciprocal Identity

Verify:

sin θ × csc θ = 1

Start with the left-hand side:

sin θ × csc θ

Replace cosecant:

= sin θ × (1/sin θ)

Cancel sin θ:

= 1

Therefore:

sin θ × csc θ = 1

The identity is verified for sin θ ≠ 0.

Verifying Identities Using Quotient Relationships

Verify:

tan θ × cos θ = sin θ

Start with the left-hand side:

tan θ × cos θ

Replace tangent:

= (sin θ/cos θ) × cos θ

Cancel cos θ:

= sin θ

Therefore:

tan θ × cos θ = sin θ

The identity is verified wherever cos θ ≠ 0.

Another Example

Verify:

cot θ × sin θ = cos θ

Start with:

cot θ × sin θ

Replace cotangent:

= (cos θ/sin θ) × sin θ

Cancel sin θ:

= cos θ

The identity is verified wherever sin θ ≠ 0.

Verifying Identities Using Pythagorean Relationships

Worked Example

Verify:

1 − sin² θ = cos² θ

Start with the fundamental identity:

sin² θ + cos² θ = 1

Subtract sin² θ from both sides:

cos² θ = 1 − sin² θ

Therefore:

1 − sin² θ = cos² θ

Worked Example

Verify:

sec² θ − tan² θ = 1

Start with:

1 + tan² θ = sec² θ

Subtract tan² θ:

sec² θ − tan² θ = 1

The identity is verified wherever cos θ ≠ 0.

Verifying More Complex Identities

More complicated expressions often require several identities.

Worked Example

Verify:

(1 − cos² θ)/sin θ = sin θ

Start with the left-hand side:

(1 − cos² θ)/sin θ

Using:

1 − cos² θ = sin² θ

Substitute:

= sin² θ/sin θ

Cancel one factor of sin θ:

= sin θ

The identity is verified wherever sin θ ≠ 0.

Worked Example

Verify:

(1 + tan² θ)cos² θ = 1

Start with:

(1 + tan² θ)cos² θ

Use:

1 + tan² θ = sec² θ

Then:

= sec² θ × cos² θ

Replace secant:

= (1/cos² θ) × cos² θ

= 1

The identity is verified wherever cos θ ≠ 0.

Simplifying Trigonometric Expressions

Simplifying a trigonometric expression means rewriting it in an equivalent form that is easier to work with.

The simplest form depends on the purpose of the calculation.

Sometimes it is useful to express everything in sine and cosine.

At other times, a reciprocal or Pythagorean identity produces a shorter result.

Worked Example: Simplify tan θ × cot θ

Use:

cot θ = 1/tan θ

Then:

tan θ × cot θ

= tan θ × (1/tan θ)

= 1

This holds where tangent and cotangent are both defined.

Worked Example: Simplify sec² θ − 1

Use:

1 + tan² θ = sec² θ

Rearrange:

sec² θ − 1 = tan² θ

Worked Example: Simplify csc² θ − cot² θ

Use:

1 + cot² θ = csc² θ

Therefore:

csc² θ − cot² θ = 1

Simplifying Expressions with Fractions

Trigonometric fractions can often be simplified by replacing functions with sine and cosine.

Worked Example

Simplify:

tan θ/sec θ

Replace tangent and secant:

= (sin θ/cos θ)/(1/cos θ)

Multiply by the reciprocal:

= (sin θ/cos θ) × cos θ

= sin θ

This is valid where cos θ ≠ 0.

Worked Example

Simplify:

cot θ/csc θ

Replace both functions:

= (cos θ/sin θ)/(1/sin θ)

Multiply by the reciprocal:

= (cos θ/sin θ) × sin θ

= cos θ

This is valid where sin θ ≠ 0.

Using Algebraic Techniques with Identities

Many trigonometric expressions can be simplified using familiar algebraic techniques.

These include:

• Factoring

• Expanding brackets

• Finding common denominators

• Difference of two squares

• Cancelling common factors

Worked Example: Difference of Squares

Simplify:

(1 − cos θ)(1 + cos θ)

Use:

(a − b)(a + b) = a² − b²

Therefore:

(1 − cos θ)(1 + cos θ)

= 1 − cos² θ

Using the Pythagorean identity:

= sin² θ

Worked Example: Simplifying a Fraction

Simplify:

sin² θ/(1 − cos θ)

Use:

sin² θ = 1 − cos² θ

Then:

= (1 − cos² θ)/(1 − cos θ)

Factor the numerator:

= [(1 − cos θ)(1 + cos θ)]/(1 − cos θ)

Cancel the common factor:

= 1 + cos θ

This simplification is valid where 1 − cos θ ≠ 0.

Rationalising Trigonometric Expressions

Sometimes a trigonometric expression contains a denominator such as:

1 + cos θ

or:

1 − sin θ

Multiplying by the conjugate can simplify the expression.

Worked Example

Simplify:

sin θ/(1 + cos θ)

Multiply numerator and denominator by:

1 − cos θ

Then:

= sin θ(1 − cos θ)/[(1 + cos θ)(1 − cos θ)]

Use the difference of squares:

= sin θ(1 − cos θ)/(1 − cos² θ)

Using:

1 − cos² θ = sin² θ

We obtain:

= sin θ(1 − cos θ)/sin² θ

Cancel one factor of sine:

= (1 − cos θ)/sin θ

This is an equivalent form wherever the original expression is defined.

This technique is useful in advanced trigonometry and calculus.

Proving Identities Step by Step

A successful identity proof should use clear mathematical reasoning.

Consider:

(sec θ − cos θ)/tan θ = sin θ

We will verify this identity.

Start with the Left-Hand Side

(sec θ − cos θ)/tan θ

Replace Secant and Tangent

= (1/cos θ − cos θ)/(sin θ/cos θ)

Combine the Numerator

1/cos θ − cos θ

= (1 − cos² θ)/cos θ

Therefore:

= [(1 − cos² θ)/cos θ]/(sin θ/cos θ)

Apply the Pythagorean Identity

Since:

1 − cos² θ = sin² θ

we obtain:

= (sin² θ/cos θ)/(sin θ/cos θ)

Simplify

= (sin² θ/cos θ) × (cos θ/sin θ)

Cancel common factors:

= sin θ

The identity is verified wherever the original left-hand side is defined, requiring sin θ ≠ 0 and cos θ ≠ 0.

Choosing the Best Identity

Different expressions suggest different approaches.

The best method is usually the one that reduces the number of different trigonometric functions or simplifies the algebra.

Using Identities to Determine Unknown Values

Trigonometric identities can help us calculate unknown function values.

Worked Example

Suppose:

cos θ = −12/13

and θ lies in Quadrant II.

Find sin θ, tan θ, and sec θ.

Find Sine

Use:

sin² θ + cos² θ = 1

Substitute:

sin² θ + (−12/13)² = 1

sin² θ + 144/169 = 1

sin² θ = 25/169

sin θ = ±5/13

Since θ lies in Quadrant II, sine is positive.

sin θ = 5/13

Find Tangent

tan θ = sin θ/cos θ

= (5/13)/(−12/13)

tan θ = −5/12

Find Secant

sec θ = 1/cos θ

= 1/(−12/13)

sec θ = −13/12

The signs agree with Quadrant II.

Identities and the Unit Circle

The unit circle provides a geometric interpretation of the fundamental identities.

For any angle θ:

x = cos θ

y = sin θ

Since every point on the unit circle satisfies:

x² + y² = 1

we obtain:

sin² θ + cos² θ = 1

The quotient identities follow from the ratio of coordinates:

tan θ = y/x

cot θ = x/y

The reciprocal identities follow from taking the reciprocals of the coordinate-based definitions.

The unit circle therefore connects all three groups of identities through a single geometric model.

Applications in Physics and Engineering

Trigonometric identities are useful whenever calculations involve angles, vectors, waves, or circular motion.

Resolving Forces

A force F acting at an angle θ can be separated into components:

Fₓ = F cos θ

Fᵧ = F sin θ

Using the Pythagorean identity:

Fₓ² + Fᵧ²

= F²cos² θ + F²sin² θ

Factor:

= F²(cos² θ + sin² θ)

Since:

cos² θ + sin² θ = 1

we obtain:

Fₓ² + Fᵧ² = F²

This shows that the magnitude of the force is consistent with the Pythagorean theorem.

Circular Motion

The position of an object moving in a circle of radius r can be written:

x = r cos θ

y = r sin θ

Squaring and adding:

x² + y² = r²(cos² θ + sin² θ)

Therefore:

x² + y² = r²

This confirms that the object remains at a constant distance r from the centre.

Waves and Oscillations

Sine and cosine functions are used to describe oscillations.

For example:

y(t) = A sin(ωt)

The corresponding cosine function is related through a phase shift.

Trigonometric identities help simplify expressions involving wave motion, energy, and oscillatory systems.

Checking Identities Numerically

Substituting an angle into both sides of an identity can help check whether a proposed relationship is plausible.

For example:

1 + tan² θ = sec² θ

At θ = 45°:

Left-hand side:

1 + tan² 45°

= 1 + 1²

= 2

Right-hand side:

sec² 45°

= (√2)²

= 2

Both sides agree.

However, checking one or several angles does not prove an identity.

A mathematical proof must establish that the relationship is true for every angle in the appropriate domain.

Numerical checks are useful for detecting errors, but algebraic reasoning is required for verification.

A Strategy for Verifying Trigonometric Identities

When verifying a trigonometric identity:

1. Examine both sides of the equation.

2. Identify the more complicated side.

3. Begin simplifying that side.

4. Apply reciprocal, quotient, or Pythagorean identities.

5. Rewrite expressions using sine and cosine when useful.

6. Apply algebraic techniques such as factoring or common denominators.

7. Simplify carefully without dividing by expressions that may be zero.

8. Continue until the expression matches the other side.

9. State any necessary domain restrictions.

10. Conclude that the identity has been verified.

Important Principle

It is generally clearer to transform one side into the other rather than manipulate both sides independently.

This makes the logical steps of the proof easier to follow.

Common Mistakes

Confusing identities with equations

An identity is true for all values in its domain. An equation may only be true for specific values.

Forgetting reciprocal relationships

Cosecant is the reciprocal of sine, not cosine.

Using the wrong quotient identity

Tangent is sine divided by cosine. Cotangent is cosine divided by sine.

Incorrectly applying the Pythagorean identity

The correct identity is:

sin² θ + cos² θ = 1

not:

sin θ + cos θ = 1

Forgetting to square the entire trigonometric value

sin² θ means (sin θ)².

Assuming sin² θ = sin(θ²)

These expressions are not equivalent.

Cancelling terms incorrectly

Cancellation applies to common factors, not individual terms separated by addition or subtraction.

Dividing by zero

Always consider where denominators may equal zero.

Forgetting quadrant signs

Taking a square root may produce positive and negative possibilities.

Using numerical examples as proof

Checking a few angles does not establish an identity.

Changing both sides without clear justification

It is usually easier to simplify one side until it matches the other.

Failing to factor expressions

Factoring can reveal Pythagorean identities and common factors.

Key Terms

Trigonometric Identity: An equation involving trigonometric functions that is true for every value where both sides are defined.

Reciprocal Identity: An identity connecting a trigonometric function to its reciprocal.

Quotient Identity: An identity expressing tangent or cotangent as a ratio of sine and cosine.

Pythagorean Identity: An identity derived from the relationship sin² θ + cos² θ = 1.

Sine: A trigonometric function representing the vertical coordinate on the unit circle.

Cosine: A trigonometric function representing the horizontal coordinate on the unit circle.

Tangent: The ratio of sine to cosine.

Cosecant: The reciprocal of sine.

Secant: The reciprocal of cosine.

Cotangent: The reciprocal of tangent.

Unit Circle: A circle of radius 1 centred at the origin.

Pythagorean Theorem: The relationship a² + b² = c² in a right triangle.

Equivalent Expressions: Expressions that have the same value wherever both are defined.

Simplification: Rewriting an expression in a more useful or less complicated form.

Verification: Demonstrating that a proposed identity is mathematically valid.

Mathematical Proof: A logical sequence of steps establishing that a statement is true.

Factorisation: Rewriting an expression as a product of factors.

Common Denominator: A shared denominator used when combining fractions.

Difference of Squares: The algebraic identity a² − b² = (a − b)(a + b).

Conjugate: An expression formed by changing the sign between two terms in a binomial.

Domain Restriction: A condition excluding input values for which an expression is undefined.

Quadrant: One of the four regions of the Cartesian coordinate plane.

Reference Angle: The acute angle between an angle's terminal side and the x-axis.

Trigonometric Expression: An expression containing one or more trigonometric functions.

Vector Component: The portion of a vector acting along a specified direction.

Periodic Function: A function whose values repeat at regular intervals.

Check Your Understanding

1. Define a trigonometric identity and explain how it differs from a trigonometric equation.

2. Write the three fundamental reciprocal identities.

3. Write the two quotient identities.

4. State the three Pythagorean identities.

5. If sin θ = 4/5, determine csc θ.

6. If cos θ = −3/7, determine sec θ.

7. Simplify:

sin θ × cot θ

8. Simplify:

tan θ/sec θ

9. Use a Pythagorean identity to simplify:

1 − cos² θ

10. Simplify:

sec² θ − tan² θ

11. Verify:

(1 − sin² θ)/cos θ = cos θ

State the domain restriction.

12. Verify:

tan θ × cos θ = sin θ

State the domain restriction.

13. Simplify:

(1 − sin θ)(1 + sin θ)

14. Given cos θ = −5/13 and θ lies in Quadrant II, determine sin θ, tan θ, and csc θ.

15. Verify the identity:

(sec θ − cos θ)/tan θ = sin θ

Show each step clearly and identify where the original expression is defined.

Key Takeaways

• Trigonometric identities are equations that are true for all angles where both sides are defined.

• The three main groups of fundamental identities are reciprocal, quotient, and Pythagorean identities.

• Reciprocal identities connect sine with cosecant, cosine with secant, and tangent with cotangent.

• The quotient identity for tangent is tan θ = sin θ/cos θ.

• The quotient identity for cotangent is cot θ = cos θ/sin θ.

• The fundamental Pythagorean identity is sin² θ + cos² θ = 1.

• The second Pythagorean identity is 1 + tan² θ = sec² θ.

• The third Pythagorean identity is 1 + cot² θ = csc² θ.

• Pythagorean identities can be rearranged to calculate unknown trigonometric values.

• Quadrant information is needed to determine the correct signs of values obtained by square roots.

• Trigonometric identities can simplify complicated expressions.

• Rewriting expressions using sine and cosine is often an effective simplification strategy.

• Algebraic techniques such as factoring and common denominators are important in identity proofs.

• A trigonometric identity can be verified by transforming one side into the other.

• Numerical substitution can help check an identity but does not constitute proof.

• Domain restrictions must be respected when dividing or cancelling expressions.

• The unit circle provides the geometric foundation for the fundamental trigonometric identities.

• Trigonometric identities are useful in vector calculations, circular motion, waves, and oscillations.

• Mastering fundamental trigonometric identities is essential for solving advanced trigonometric equations and developing the algebraic skills required in calculus.