The Central Idea
An identity is an equality that is true for every value in the common domain of the two expressions.
For example,
\[ \boxed{\sin^2\theta+\cos^2\theta=1} \]is true for every real \(\theta\).
Similarly,
\[ \boxed{\tan\theta=\frac{\sin\theta}{\cos\theta}} \]is true whenever \(\cos\theta\neq0\).
Identities allow one trigonometric expression to be replaced by another equivalent expression.
Identity versus Equation
An identity is different from an equation that is true only for selected inputs.
For example,
\[ \sin x=0 \]is not an identity because it is true only at certain values of \(x\).
But
\[ \sin^2x+\cos^2x=1 \]is an identity because it is true throughout its domain.
Conditional Equality
Some equalities are true only under a condition.
For example,
\[ \sin x=\cos x \]is true for certain angles but not for all angles.
Such an equation should be solved for its valid inputs rather than treated as an identity.
The Reciprocal Identities
The reciprocal pairs give
\[ \boxed{\csc\theta=\frac1{\sin\theta}}, \] \[ \boxed{\sec\theta=\frac1{\cos\theta}}, \]and
\[ \boxed{\cot\theta=\frac1{\tan\theta}}. \]These identities apply wherever the denominators are nonzero.
Reciprocal Identities in Reverse
The same relationships can be written in reverse:
\[ \boxed{\sin\theta=\frac1{\csc\theta}}, \] \[ \boxed{\cos\theta=\frac1{\sec\theta}}, \]and
\[ \boxed{\tan\theta=\frac1{\cot\theta}}. \]Which direction is more useful depends on the expression being simplified.
Reciprocal Product Identities
Multiplying reciprocal partners gives
\[ \boxed{\sin\theta\csc\theta=1}, \] \[ \boxed{\cos\theta\sec\theta=1}, \]and
\[ \boxed{\tan\theta\cot\theta=1}, \]wherever both functions are defined.
The Quotient Identities
Tangent and cotangent can be expressed entirely in terms of sine and cosine:
\[ \boxed{\tan\theta=\frac{\sin\theta}{\cos\theta}} \]and
\[ \boxed{\cot\theta=\frac{\cos\theta}{\sin\theta}}. \]These identities are especially useful when an expression contains several different trigonometric functions.
Deriving the Tangent Quotient Identity
In a right triangle,
\[ \sin\theta=\frac{O}{H} \]and
\[ \cos\theta=\frac{A}{H}. \]Divide:
\[ \frac{\sin\theta}{\cos\theta} = \frac{O/H}{A/H} = \frac{O}{A}. \]Since
\[ \tan\theta=\frac{O}{A}, \]we obtain
\[ \boxed{\tan\theta=\frac{\sin\theta}{\cos\theta}}. \]Deriving the Cotangent Quotient Identity
Taking the reciprocal of the tangent identity gives
\[ \frac1{\tan\theta} = \frac{\cos\theta}{\sin\theta}. \]Since
\[ \frac1{\tan\theta}=\cot\theta, \]we obtain
\[ \boxed{\cot\theta=\frac{\cos\theta}{\sin\theta}}. \]The Fundamental Pythagorean Identity
The most important trigonometric identity is
\[ \boxed{\sin^2\theta+\cos^2\theta=1}. \]It comes directly from the unit-circle equation
\[ x^2+y^2=1 \]after substituting
\[ x=\cos\theta \]and
\[ y=\sin\theta. \]Rearranging the Fundamental Identity
From
\[ \sin^2\theta+\cos^2\theta=1, \]we can isolate either squared function:
\[ \boxed{\sin^2\theta=1-\cos^2\theta} \]and
\[ \boxed{\cos^2\theta=1-\sin^2\theta}. \]These forms are often more useful during simplification.
The Tangent–Secant Pythagorean Identity
Divide
\[ \sin^2\theta+\cos^2\theta=1 \]by \(\cos^2\theta\), assuming \(\cos\theta\neq0\):
\[ \frac{\sin^2\theta}{\cos^2\theta} + \frac{\cos^2\theta}{\cos^2\theta} = \frac1{\cos^2\theta}. \]Therefore,
\[ \boxed{\tan^2\theta+1=\sec^2\theta}. \]Useful Rearrangements of the Tangent–Secant Identity
From
\[ \tan^2\theta+1=\sec^2\theta, \]we obtain
\[ \boxed{\tan^2\theta=\sec^2\theta-1} \]and
\[ \boxed{1=\sec^2\theta-\tan^2\theta}. \]The Cotangent–Cosecant Pythagorean Identity
Divide
\[ \sin^2\theta+\cos^2\theta=1 \]by \(\sin^2\theta\), assuming \(\sin\theta\neq0\):
\[ 1+\frac{\cos^2\theta}{\sin^2\theta} = \frac1{\sin^2\theta}. \]Therefore,
\[ \boxed{1+\cot^2\theta=\csc^2\theta}. \]Useful Rearrangements of the Cotangent–Cosecant Identity
From
\[ 1+\cot^2\theta=\csc^2\theta, \]we obtain
\[ \boxed{\cot^2\theta=\csc^2\theta-1} \]and
\[ \boxed{1=\csc^2\theta-\cot^2\theta}. \]The Three Pythagorean Identities
The family can be summarized as
\[ \boxed{\sin^2\theta+\cos^2\theta=1}, \] \[ \boxed{1+\tan^2\theta=\sec^2\theta}, \]and
\[ \boxed{1+\cot^2\theta=\csc^2\theta}. \]The latter two are consequences of the first.
Even–Odd Identities
Unit-circle symmetry gives
\[ \boxed{\sin(-\theta)=-\sin\theta}, \] \[ \boxed{\cos(-\theta)=\cos\theta}, \]and
\[ \boxed{\tan(-\theta)=-\tan\theta}. \]Sine and tangent are odd functions, while cosine is even.
Even–Odd Identities for Reciprocal Functions
Reciprocal partners inherit the same parity:
\[ \boxed{\csc(-\theta)=-\csc\theta}, \] \[ \boxed{\sec(-\theta)=\sec\theta}, \]and
\[ \boxed{\cot(-\theta)=-\cot\theta}. \]Cofunction Identities
Complementary angles exchange opposite and adjacent side roles.
Therefore,
\[ \boxed{ \sin\left(\frac{\pi}{2}-\theta\right)=\cos\theta } \]and
\[ \boxed{ \cos\left(\frac{\pi}{2}-\theta\right)=\sin\theta }. \]Tangent and Cotangent Cofunction Identities
The complementary relationship also gives
\[ \boxed{ \tan\left(\frac{\pi}{2}-\theta\right)=\cot\theta } \]and
\[ \boxed{ \cot\left(\frac{\pi}{2}-\theta\right)=\tan\theta }. \]Secant and Cosecant Cofunction Identities
Taking reciprocals of the sine–cosine cofunction identities gives
\[ \boxed{ \sec\left(\frac{\pi}{2}-\theta\right)=\csc\theta } \]and
\[ \boxed{ \csc\left(\frac{\pi}{2}-\theta\right)=\sec\theta }. \]Periodicity Identities
Sine and cosine repeat every full revolution:
\[ \boxed{\sin(\theta+2\pi)=\sin\theta}, \] \[ \boxed{\cos(\theta+2\pi)=\cos\theta}. \]Tangent and cotangent repeat every half revolution:
\[ \boxed{\tan(\theta+\pi)=\tan\theta}, \] \[ \boxed{\cot(\theta+\pi)=\cot\theta}. \]Why Identity Verification Is Different from Solving
When verifying an identity, the goal is not to find values of the variable.
The goal is to transform one side into an equivalent form until it matches the other side.
For example, to verify
\[ \frac{\sin\theta}{\cos\theta}=\tan\theta, \]we recognize the quotient identity rather than solving for \(\theta\).
Work on One Side at a Time
A reliable identity-verification habit is to begin with the more complicated side and rewrite it until it becomes the simpler side.
This keeps the argument logically clear.
Avoid changing both sides independently in a way that hides whether each transformation preserves equivalence.
Example: Verify a Reciprocal Identity
Verify
\[ \sin\theta\csc\theta=1. \]Begin with the left side:
\[ \sin\theta\csc\theta. \]Replace cosecant:
\[ \sin\theta\left(\frac1{\sin\theta}\right). \]Therefore,
\[ \boxed{\sin\theta\csc\theta=1} \]wherever \(\sin\theta\neq0\).
Example: Verify a Quotient Identity
Verify
\[ \frac{\tan\theta}{\sec\theta}=\sin\theta. \]Begin with the left side:
\[ \frac{\tan\theta}{\sec\theta}. \]Rewrite in sine and cosine:
\[ \frac{\sin\theta/\cos\theta}{1/\cos\theta}. \]Dividing by a fraction means multiplying by its reciprocal:
\[ \frac{\sin\theta}{\cos\theta}\cdot\cos\theta. \]Therefore,
\[ \boxed{\frac{\tan\theta}{\sec\theta}=\sin\theta}. \]Example: Use a Pythagorean Identity
Simplify
\[ 1-\sin^2\theta. \]From
\[ \sin^2\theta+\cos^2\theta=1, \]we have
\[ \boxed{1-\sin^2\theta=\cos^2\theta}. \]Example: Simplify with Tangent and Secant
Simplify
\[ \sec^2\theta-\tan^2\theta. \]Rearranging
\[ 1+\tan^2\theta=\sec^2\theta \]gives
\[ \boxed{\sec^2\theta-\tan^2\theta=1}. \]Example: Simplify with Cotangent and Cosecant
Simplify
\[ \csc^2\theta-\cot^2\theta. \]From
\[ 1+\cot^2\theta=\csc^2\theta, \]we obtain
\[ \boxed{\csc^2\theta-\cot^2\theta=1}. \]Convert to Sine and Cosine
When an expression contains several different trigonometric functions, converting everything to sine and cosine often reveals common factors or cancellations.
The main replacements are
\[ \tan\theta=\frac{\sin\theta}{\cos\theta}, \qquad \cot\theta=\frac{\cos\theta}{\sin\theta}, \] \[ \sec\theta=\frac1{\cos\theta}, \qquad \csc\theta=\frac1{\sin\theta}. \]Example: Convert Everything to Sine and Cosine
Simplify
\[ \tan\theta\cos\theta. \]Replace tangent:
\[ \frac{\sin\theta}{\cos\theta}\cos\theta. \]Cancel the common factor where valid:
\[ \boxed{\tan\theta\cos\theta=\sin\theta}. \]Example: Reciprocal Cancellation
Simplify
\[ \frac{\cos\theta}{\sec\theta}. \]Since
\[ \sec\theta=\frac1{\cos\theta}, \]we have
\[ \frac{\cos\theta}{1/\cos\theta} = \cos^2\theta. \]Therefore,
\[ \boxed{\frac{\cos\theta}{\sec\theta}=\cos^2\theta}. \]Factoring Trigonometric Expressions
Ordinary algebraic factoring applies to trigonometric expressions.
For example,
\[ \sin\theta+\sin\theta\tan^2\theta \]has the common factor \(\sin\theta\):
\[ \sin\theta(1+\tan^2\theta). \]Use the Pythagorean identity:
\[ \boxed{ \sin\theta+\sin\theta\tan^2\theta = \sin\theta\sec^2\theta }. \]Combining Fractions
Trigonometric expressions follow ordinary fraction rules.
Consider
\[ \frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta}. \]Use the common denominator
\[ \sin\theta\cos\theta. \]Then
\[ \frac{\sin^2\theta+\cos^2\theta} {\sin\theta\cos\theta}. \]Apply the Pythagorean identity:
\[ \boxed{ \frac{1}{\sin\theta\cos\theta} }. \]Recognizing a Hidden Pythagorean Numerator
In the expression
\[ \frac{\sin^2\theta+\cos^2\theta}{\cos\theta}, \]the numerator equals \(1\).
Therefore,
\[ \frac1{\cos\theta} = \boxed{\sec\theta}. \]Multiplying by a Conjugate
Some trigonometric expressions contain sums or differences such as
\[ 1-\cos\theta. \]Multiplying by the conjugate
\[ 1+\cos\theta \]can create a Pythagorean expression:
\[ (1-\cos\theta)(1+\cos\theta) = 1-\cos^2\theta = \sin^2\theta. \]This technique becomes useful in more advanced identity manipulations.
Example: Simplify with a Conjugate
Consider
\[ \frac{\sin^2\theta}{1-\cos\theta}. \]Since
\[ \sin^2\theta=1-\cos^2\theta, \]factor the numerator:
\[ \frac{(1-\cos\theta)(1+\cos\theta)} {1-\cos\theta}. \]Where the canceled factor is nonzero,
\[ \boxed{ \frac{\sin^2\theta}{1-\cos\theta} = 1+\cos\theta }. \]Domain Restrictions Matter
Algebraic simplification can hide values that were excluded from the original expression.
For example,
\[ \frac{\sin^2\theta}{1-\cos\theta} \]is undefined when
\[ 1-\cos\theta=0. \]Thus,
\[ \cos\theta=1 \]is excluded from the original domain even though
\[ 1+\cos\theta \]itself is defined there.
Equivalent Expressions Can Have Different Written Domains
Simplification preserves values on the original domain, but canceling a factor can make the simplified expression appear defined at additional inputs.
Therefore, when an identity involves quotients, the precise statement is understood over the common domain where both sides are defined.
Numerical Checks Are Useful but Not Proof
Substituting several values can help detect an incorrect identity.
For example, if someone claims
\[ \sin\theta+\cos\theta=1, \]try
\[ \theta=\frac{\pi}{4}. \]Then
\[ \frac{\sqrt2}{2}+\frac{\sqrt2}{2} = \sqrt2 \neq1. \]One counterexample disproves the proposed identity.
Why Numerical Agreement Is Not Enough
If two expressions agree at one, ten, or even a thousand tested values, that does not by itself prove they agree for every input.
Identity verification requires a general algebraic or geometric argument.
Numerical testing is best used as a secondary check.
Example: Verify an Even–Odd Identity
Verify
\[ \sec(-\theta)=\sec\theta. \]Begin with the left side:
\[ \sec(-\theta) = \frac1{\cos(-\theta)}. \]Since cosine is even,
\[ \cos(-\theta)=\cos\theta. \]Therefore,
\[ \boxed{\sec(-\theta)=\sec\theta}. \]Example: Verify a Cofunction Identity
Verify
\[ \csc\left(\frac{\pi}{2}-\theta\right)=\sec\theta. \]Begin with the left side:
\[ \csc\left(\frac{\pi}{2}-\theta\right) = \frac1{ \sin\left(\frac{\pi}{2}-\theta\right) }. \]Use
\[ \sin\left(\frac{\pi}{2}-\theta\right)=\cos\theta. \]Thus,
\[ \frac1{\cos\theta} = \boxed{\sec\theta}. \]Example: A Multi-Step Verification
Verify
\[ \frac{1-\cos^2\theta}{\sin\theta} = \sin\theta. \]Begin with the left side:
\[ \frac{1-\cos^2\theta}{\sin\theta}. \]Use
\[ 1-\cos^2\theta=\sin^2\theta. \]Then
\[ \frac{\sin^2\theta}{\sin\theta} = \boxed{\sin\theta}, \]for inputs where the original denominator is nonzero.
Example: Simplify a Mixed Expression
Simplify
\[ \frac{\sec\theta-\cos\theta}{\tan\theta}. \]Rewrite secant and tangent:
\[ \frac{ \frac1{\cos\theta}-\cos\theta }{ \frac{\sin\theta}{\cos\theta} }. \]Combine the numerator:
\[ \frac{ \frac{1-\cos^2\theta}{\cos\theta} }{ \frac{\sin\theta}{\cos\theta} }. \]Use
\[ 1-\cos^2\theta=\sin^2\theta. \]Then
\[ \frac{ \sin^2\theta/\cos\theta }{ \sin\theta/\cos\theta } = \boxed{\sin\theta}. \]Choose the Identity That Matches the Structure
Effective simplification is largely pattern recognition.
If you see
\[ 1-\sin^2\theta, \]think
\[ \cos^2\theta. \]If you see
\[ 1+\tan^2\theta, \]think
\[ \sec^2\theta. \]If you see several different functions, consider rewriting them in sine and cosine.
Do Not Expand without a Purpose
Algebraic expansion can sometimes make an identity harder to recognize.
Before expanding, ask whether factoring, a reciprocal identity, a quotient identity, or a Pythagorean replacement exposes structure more directly.
Factoring Can Reveal an Identity
Consider
\[ \cos^3\theta+\cos\theta\sin^2\theta. \]Factor \(\cos\theta\):
\[ \cos\theta(\cos^2\theta+\sin^2\theta). \]Use the Pythagorean identity:
\[ \boxed{ \cos^3\theta+\cos\theta\sin^2\theta = \cos\theta }. \]A Common-Denominator Example
Simplify
\[ \frac1{1+\sin\theta} + \frac1{1-\sin\theta}. \]Use the common denominator
\[ (1+\sin\theta)(1-\sin\theta) = 1-\sin^2\theta. \]The numerator becomes
\[ (1-\sin\theta)+(1+\sin\theta)=2. \]Therefore,
\[ \frac{2}{1-\sin^2\theta} = \frac{2}{\cos^2\theta} = \boxed{2\sec^2\theta}. \]Identity Chains
One simplification may use several identities in sequence.
For example,
\[ \frac{\tan\theta}{\sec\theta} \]first becomes
\[ \frac{\sin\theta/\cos\theta}{1/\cos\theta}, \]then simplifies to
\[ \sin\theta. \]Each step should preserve equivalence over the relevant domain.
Identity Verification Is Algebra with Trigonometric Vocabulary
The underlying algebra remains familiar:
- factor common terms;
- combine fractions;
- multiply by conjugates;
- cancel common factors when valid;
- substitute equivalent expressions;
- track domain restrictions.
The new skill is recognizing which trigonometric relationship makes the algebra simpler.
Common Error: Trying to Prove an Identity by Checking One Angle
Showing that both sides have the same value at
\[ \theta=30^\circ \]verifies only that single input.
An identity requires a general argument valid throughout the common domain.
Common Error: Changing Both Sides Randomly
If both sides are rewritten independently, it can become unclear whether the transformations actually connect the original expressions.
A cleaner strategy is usually:
- start with the more complicated side;
- apply valid identities and algebra;
- stop when the other side appears.
Common Error: Canceling Terms across Addition
In
\[ \frac{\sin\theta+\cos\theta}{\sin\theta}, \]you cannot cancel \(\sin\theta\) from only the first term in the numerator.
Instead, split the fraction:
\[ \frac{\sin\theta}{\sin\theta} + \frac{\cos\theta}{\sin\theta} = 1+\cot\theta. \]Cancellation applies to factors, not isolated terms within a sum.
Common Error: Forgetting Function Squares
The identity
\[ 1+\tan^2\theta=\sec^2\theta \]does not imply
\[ 1+\tan\theta=\sec\theta. \]The squares are essential.
Common Error: Taking Square Roots without Sign Information
From
\[ \sin^2\theta=1-\cos^2\theta, \]it does not follow that
\[ \sin\theta=\sqrt{1-\cos^2\theta} \]for every angle.
In general,
\[ \boxed{ \sin\theta=\pm\sqrt{1-\cos^2\theta} } \]with the sign determined by the angle or other given information.
Common Error: Ignoring Original Restrictions after Cancellation
If a denominator contains
\[ \sin\theta, \]then inputs with
\[ \sin\theta=0 \]are excluded from the original expression.
Canceling the factor later does not restore those values to the original domain.
Common Error: Confusing Identity Verification with Equation Solving
If the task is to verify
\[ 1-\sin^2\theta=\cos^2\theta, \]do not try to isolate \(\theta\).
The equality is already intended as a general relationship.
The next lesson will focus specifically on equations whose valid angles must be found.
A Reliable Procedure for Verifying an Identity
- Identify the more complicated side.
- Look for reciprocal, quotient, or Pythagorean patterns.
- Convert to sine and cosine if several functions are mixed.
- Apply ordinary algebra carefully.
- Factor or combine fractions when useful.
- Use conjugates when a difference of squares may appear.
- Track all denominator restrictions.
- Continue until the expression matches the other side.
- Use a numerical check only as secondary verification.
A Reliable Procedure for Simplifying a Trigonometric Expression
- Inventory the functions present.
- Choose a target form if one is suggested by the problem.
- Replace reciprocal or quotient functions when useful.
- Search for one of the three Pythagorean identities.
- Factor common expressions.
- Combine fractions using a common denominator.
- Cancel only common factors, never separate terms.
- State or preserve relevant domain restrictions.
Practice 1: Reciprocal Identity
Simplify:
\[ \cos\theta\sec\theta. \]Practice 2: Quotient Identity
Simplify:
\[ \frac{\sin\theta}{\cos\theta}. \]Practice 3: Pythagorean Identity
Simplify:
\[ 1-\cos^2\theta. \]Practice 4: Tangent–Secant Identity
Simplify:
\[ \sec^2\theta-1. \]Practice 5: Cotangent–Cosecant Identity
Simplify:
\[ \csc^2\theta-1. \]Practice 6: Mixed Functions
Simplify:
\[ \frac{\tan\theta}{\sec\theta}. \]Practice 7: Factor First
Simplify:
\[ \sin^3\theta+\sin\theta\cos^2\theta. \]Practice 8: Common Denominator
Simplify:
\[ \frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta}. \]Practice 9: Even–Odd Identity
Simplify:
\[ \cos(-\theta). \]Practice 10: Cofunction Identity
Simplify:
\[ \sin\left(\frac{\pi}{2}-\theta\right). \]Practice 11: Verify an Identity
Verify:
\[ \frac{1-\sin^2\theta}{\cos\theta} = \cos\theta. \]Practice 12: Identify the Error
A student claims that
\[ 1+\tan\theta=\sec\theta \]follows from the Pythagorean identities.
Explain the error.
Practice Answers
Practice 1
\[ \cos\theta\sec\theta = \cos\theta\left(\frac1{\cos\theta}\right) = \boxed{1}, \]where the original expression is defined.
Practice 2
\[ \boxed{ \frac{\sin\theta}{\cos\theta} = \tan\theta }. \]Practice 3
\[ \boxed{ 1-\cos^2\theta = \sin^2\theta }. \]Practice 4
\[ \boxed{ \sec^2\theta-1 = \tan^2\theta }. \]Practice 5
\[ \boxed{ \csc^2\theta-1 = \cot^2\theta }. \]Practice 6
\[ \frac{\tan\theta}{\sec\theta} = \frac{\sin\theta/\cos\theta}{1/\cos\theta} = \boxed{\sin\theta}. \]Practice 7
Factor \(\sin\theta\):
\[ \sin\theta(\sin^2\theta+\cos^2\theta). \]Therefore,
\[ \boxed{ \sin^3\theta+\sin\theta\cos^2\theta = \sin\theta }. \]Practice 8
\[ \frac{\sin^2\theta+\cos^2\theta} {\sin\theta\cos\theta} = \boxed{ \frac1{\sin\theta\cos\theta} }. \]An equivalent form is
\[ \boxed{\sec\theta\csc\theta}. \]Practice 9
Cosine is even:
\[ \boxed{\cos(-\theta)=\cos\theta}. \]Practice 10
\[ \boxed{ \sin\left(\frac{\pi}{2}-\theta\right) = \cos\theta }. \]Practice 11
Begin with the left side:
\[ \frac{1-\sin^2\theta}{\cos\theta}. \]Since
\[ 1-\sin^2\theta=\cos^2\theta, \]we obtain
\[ \frac{\cos^2\theta}{\cos\theta} = \boxed{\cos\theta}, \]for inputs where the original denominator is nonzero.
Practice 12
The correct Pythagorean identity is
\[ \boxed{1+\tan^2\theta=\sec^2\theta}. \]The squares cannot be removed term by term.
In general,
\[ 1+\tan\theta\neq\sec\theta. \]What to Remember
The major identity families are:
Reciprocal:
\[ \boxed{ \csc\theta=\frac1{\sin\theta}, \qquad \sec\theta=\frac1{\cos\theta}, \qquad \cot\theta=\frac1{\tan\theta} }. \]Quotient:
\[ \boxed{ \tan\theta=\frac{\sin\theta}{\cos\theta}, \qquad \cot\theta=\frac{\cos\theta}{\sin\theta} }. \]Pythagorean:
\[ \boxed{\sin^2\theta+\cos^2\theta=1}, \] \[ \boxed{1+\tan^2\theta=\sec^2\theta}, \] \[ \boxed{1+\cot^2\theta=\csc^2\theta}. \]- Identities are true throughout their common domains.
- Identity verification is not the same task as equation solving.
- Converting mixed expressions to sine and cosine often simplifies them.
- Ordinary algebraic tools—factoring, common denominators, conjugates, and cancellation—still apply.
- Cancellation applies to factors, not individual terms in sums.
- Domain restrictions must be preserved when denominators or canceled factors are involved.
- Numerical checks can disprove a false identity but do not establish a general proof.
- Even–odd, cofunction, and periodic identities encode geometric symmetries of the unit circle.
Where We Go Next
Trigonometric identities provide tools for rewriting expressions without changing their values.
The next lesson, Trigonometric Equations, uses these identities together with unit-circle values, inverse functions, factoring, and periodicity to find the angles that satisfy particular trigonometric conditions.