Prove trigonometric identities using the course identity families

Alberta Grade 12 Mathematics

Trigonometry

Using familiar identity families to show two expressions are equal

An identity is an equation that is true for every angle in its domain. The domain is the set of angle values for which the expressions are defined. To prove an identity, use known relationships to change one side until it matches the other. You do not need to test every angle. You do need to preserve equality at every step and avoid values that make a denominator zero.

What you will learn

Distinguish an identity from an equation that is true only for certain angles.
Recall the reciprocal, quotient, and Pythagorean identity families.
Prove identities by changing one side at a time and explaining each change.
State the angle restrictions that apply to a proof.

1. Review: expressions, equations, and restrictions

An expression is a mathematical phrase, such as a quotient of trigonometric ratios. An equation says that two expressions are equal. An identity is an equation that holds for all permitted values of its variable.

The trigonometric ratios may be written as sine, cosine, and tangent, or as their abbreviations. The quotient identities connect tangent to sine and cosine. Since division by zero is undefined, tangent is undefined when cosine is zero.

When a proof contains a denominator, note where it can be zero. A proof that divides by an expression is valid only where that expression is nonzero. Restrictions are part of the identity's domain, not a minor detail.

\tan x=\frac{\sin x}{\cos x}

An identity is true for every angle in its domain.
Never divide by zero.
Keep track of values excluded by denominators.

2. The identity families

The reciprocal identities relate each ratio to its reciprocal. The quotient identities express tangent and cotangent using sine and cosine. The Pythagorean identities come from the basic relation between sine and cosine and can be rearranged to isolate different squares.

A reciprocal identity is useful when an expression contains a reciprocal ratio. A quotient identity is useful when an expression mixes tangent with sine or cosine. A Pythagorean identity is useful when a proof contains squared sine or cosine terms.

You may also use sum-and-difference and double-angle identities when they are part of the course identity families and suit the expression. Choose a relationship that makes the expression simpler. Do not use an identity merely because it appears on a formula sheet.

\csc x=\frac{1}{\sin x},\quad \sec x=\frac{1}{\cos x},\quad \cot x=\frac{\cos x}{\sin x},\quad \sin^2x+\cos^2x=1,\quad 1+\tan^2x=\sec^2x

Reciprocal identities: cosecant, secant, and cotangent are reciprocals of sine, cosine, and tangent.
Quotient identities rewrite tangent and cotangent with sine and cosine.
Pythagorean identities connect squared trigonometric ratios.
An identity may be rearranged using ordinary algebra.

3. A reliable proof method

Begin by identifying the more complicated side. Work on that side alone. Replacing one side at a time makes it easier to see why each line follows from the previous one.

Look for a useful identity, then simplify with algebra. Factoring, finding a common denominator, and cancelling a common nonzero factor are ordinary algebra steps. State a reason when a step might not be obvious.

Stop when the expression matches the untouched side. If the two sides have different forms, rewrite one in a form that can be compared with the other. A numerical check can help you notice an error, but it cannot prove an identity.

Work on one side at a time.
Choose identities to simplify the expression, not to make it longer.
Explain substitutions and algebra steps.
Check restrictions before cancelling or dividing.

4. Keep the domain visible

A denominator restriction follows from the original expression. For example, an expression containing tangent requires cosine to be nonzero because tangent is sine divided by cosine. An expression containing cotangent requires sine to be nonzero.

A proof may simplify to an expression that is defined at additional angles. That does not restore angles excluded from the original expression. Report the domain of the original identity.

Exact identities describe equality, not an approximate match. A calculator can display rounded values, so decimal agreement at a few angles is not enough to establish an identity.

Restrictions come from the original expressions.
Simplification does not add excluded angles back into the domain.
Use algebraic identities for proof; use decimal checks only as a check.

Worked example

Example 1: Use a quotient identity and a Pythagorean identity

Prove that

\tan x\cos x=\sin x

for all angles where the left side is defined.

Unit-circle ratios

A reference view of sine and cosine ratios; the proof itself is algebraic.

State the restriction
The left side contains tangent, so cosine must be nonzero. This excludes angles where cosine equals zero.
$\cos x\ne 0$
Replace tangent
Use the quotient identity to express tangent as sine divided by cosine. The restriction ensures this denominator is nonzero.
$\tan x\cos x=\frac{\sin x}{\cos x}\cos x$
Simplify
The cosine factors cancel because cosine is nonzero on the stated domain. The result is the right side, so the identity is proved there.
$\frac{\sin x}{\cos x}\cos x=\sin x$

Answer: The identity is true for all

x

such that

\cos x\ne0

.

Check: The proof began with the side containing tangent and used an identity valid wherever tangent is defined.

Worked example

Example 2: Simplify a squared expression

Prove that

\frac{1-\cos^2x}{\sin x}=\sin x

for angles where the left side is defined.

Sine and cosine relationship

The unit-circle relationship motivates the Pythagorean identity used in the algebraic proof.

State the restriction
The denominator is sine, so the original expression requires sine to be nonzero.
$\sin x\ne0$
Use the Pythagorean identity
Rearrange the identity $\sin^2x+\cos^2x=1$ to replace the numerator. This substitution is valid for every angle.
$1-\cos^2x=\sin^2x$
Reduce the quotient
Substitute the squared sine into the numerator, then cancel one sine factor. The restriction makes the cancellation valid.
$\frac{1-\cos^2x}{\sin x}=\frac{\sin^2x}{\sin x}=\sin x$

Answer: The identity holds for all

x

such that

\sin x\ne0

.

Check: The simplified right side may be defined when sine is zero, but the original fraction is not. Keep the original restriction.

Worked example

Example 3: Combine reciprocal and Pythagorean identities

Prove that

\sec^2x-1=\tan^2x

for all angles where the expressions are defined.

Trigonometric ratio relationships

A reference view of the sine and cosine ratios related by the Pythagorean identity.

State the restriction
Secant is the reciprocal of cosine, so the expressions require cosine to be nonzero. Tangent has the same restriction.
$\cos x\ne0$
Rewrite secant
Use the reciprocal identity for secant. Squaring gives a squared reciprocal.
$\sec^2x-1=\frac{1}{\cos^2x}-1$
Combine terms
Write one as a fraction with denominator cosine squared. Subtract the numerators, then use the Pythagorean identity.
$\frac{1}{\cos^2x}-1=\frac{1-\cos^2x}{\cos^2x}=\frac{\sin^2x}{\cos^2x}$
Use the quotient identity
The last quotient is the square of sine divided by cosine, which is tangent squared. This matches the required side.
$\frac{\sin^2x}{\cos^2x}=\tan^2x$

Answer: The identity holds for all

x

such that

\cos x\ne0

.

Check: The proof uses only the reciprocal, Pythagorean, and quotient identities, with the denominator restriction maintained.

Common mistakes and how to avoid them

Changing both sides at once without showing the connection.

Correction: Work on one side at a time and state the identity or algebra rule used.

Cancelling a trigonometric factor without checking whether it is zero.

Correction: State the restriction first. A factor may be cancelled only where it is nonzero.

Treating a few matching calculator values as proof.

Correction: Use identity families and algebra to establish equality for every permitted angle.

Forgetting that simplification can hide original restrictions.

Correction: Determine the domain from the original expressions and keep it with the conclusion.

Lesson summary

An identity is true for every angle in its domain.
Use reciprocal, quotient, and Pythagorean identities to rewrite expressions.
Work on one side at a time and justify each change.
Check denominators and retain restrictions from the original statement.

Check your understanding

Question 1

Which identity directly rewrites

\tan x

using sine and cosine?

$\tan x=\frac{\sin x}{\cos x}$
$\tan x=\frac{\cos x}{\sin x}$
$\tan x=\sin x+\cos x$
$\tan x=\frac{1}{\sin x}$

Show answer and explanation

\tan x=\frac{\sin x}{\cos x}

The quotient identity defines tangent as sine divided by cosine. Cosine must be nonzero.

Question 2

What restriction is required for

\frac{1-\sin^2x}{\cos x}

?

$\sin x\ne0$
$\cos x\ne0$
$\tan x\ne0$
There is no restriction.

Show answer and explanation

\cos x\ne0

Cosine is the denominator, so it cannot equal zero.

Question 3

Which identity can replace

1-\sin^2x

?

$\cos^2x$
$\sin x\cos x$
$1+\cos^2x$
$\tan^2x$

Show answer and explanation

\cos^2x

Rearranging

\sin^2x+\cos^2x=1

gives

1-\sin^2x=\cos^2x

.

Key terms

Identity: An equation that is true for every permitted value of its variable.
Domain: The set of input values for which an expression is defined.
Reciprocal identity: A relationship that expresses one trigonometric ratio as the reciprocal of another.
Quotient identity: A relationship that expresses tangent or cotangent as a quotient of sine and cosine.
Pythagorean identity: A trigonometric relationship based on $\sin^2x+\cos^2x=1$ and its rearrangements.

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Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-1), expectation TR6. It is a study resource, not an official curriculum publication.

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TR6 · Prove trigonometric identities using the course identity families

Alberta Grade 12 Mathematics

Using familiar identity families to show two expressions are equal

What you will learn

1. Review: expressions, equations, and restrictions

2. The identity families

3. A reliable proof method

4. Keep the domain visible

Example 1: Use a quotient identity and a Pythagorean identity

Example 2: Simplify a squared expression

Example 3: Combine reciprocal and Pythagorean identities

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Math 30-1

About this lesson and its review