B3.3 Prove trigonometric identities | Ontario Grade 12 Mathematics

Ontario Grade 12 Mathematics

Trigonometric Functions

A Grade 12 guide to using known relationships to show two expressions are equal

A trigonometric identity is an equality that is true for every allowed input, not just for one angle. To prove one, start with one side and use known trigonometric relationships to rewrite it until it matches the other side. Each rewrite must preserve the value of the expression wherever it is defined. This lesson reviews the relationships needed for that work, then applies them in a guided proof.

What you will learn

Explain what a trigonometric identity is and how it differs from an equation that is true only for certain values.
Recall the reciprocal, quotient, and Pythagorean trigonometric identities used in proofs.
Prove an identity by changing one side into the other while keeping each step equivalent.
Recognize common errors, including changing both sides without showing why and ignoring where expressions are undefined.

1. Prerequisite bridge: equations, ratios, and domains

An equation states that two expressions have the same value. Some equations are true only for particular values of the variable. For example,

x+2=5

is true when

x=3

. An identity is stronger: it is true for every allowed value of its variable.

The word allowed matters. Trigonometric expressions can be undefined at some angles. For example,

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

is defined only when

\cos x\ne 0

. A proof must concern values for which the expressions being compared are defined.

When simplifying, use familiar algebra carefully. You may factor, combine fractions, or replace a quantity with an equal one. You may cancel a common factor only when that factor is not zero. These rules apply to trigonometric expressions just as they do to ordinary algebra.

\tan x=\frac{\sin x}{\cos x},\quad \cos x\ne 0

An identity is true for all inputs where both sides are defined.
Check denominators and excluded values before cancelling or dividing.
A proof gives a chain of equal expressions, not just a numerical example.

2. The main relationships and how to choose one

The sine, cosine, and tangent ratios are connected by standard identities. The reciprocal identities define cosecant, secant, and cotangent as reciprocals. The quotient identities express tangent and cotangent as ratios of sine and cosine. The Pythagorean identities connect squares of these ratios.

In a proof, first look for a form that appears in the expression you need to change. A tangent may be easier to work with as sine divided by cosine. A squared sine beside a squared cosine may suggest using the Pythagorean identity. Rewriting is useful only when it moves the expression toward the target.

A reliable approach is to work on one side, usually the more complicated one. Write the identity you use at the step where it applies. Do not assume that two sides are equal just because a calculator gives matching decimal values for a few angles. A few checks can find errors, but they do not establish an identity.

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

Reciprocal identities connect a ratio with its reciprocal.
Quotient identities can replace tangent or cotangent with sine and cosine.
Pythagorean identities often simplify sums or differences of squared ratios.
Choose substitutions that make the current expression resemble the other side.

3. A proof as a chain of equivalent expressions

To prove an identity, write one side and transform it step by step. Each line should have the same value as the line before it for all relevant inputs. The final line should be exactly the other side, or a form that is clearly equal to it.

The choice of starting side is strategic. If one side contains more fractions or trigonometric functions, it often offers more opportunities to simplify. Starting with the simpler side can also work, but do not move terms across an equality as if you were solving for an unknown. The goal is to show that a single expression can be rewritten into the target.

For a proof that contains fractions, keep track of where each denominator is zero. If the original expression is undefined at an angle, the identity does not claim that the expression has a value there. Avoid creating a proof that silently divides by a quantity that could be zero.

A short note beside a line can name the rule used, such as a quotient identity or a difference of squares. The algebra itself should still be visible. This lets a reader check both the choice of identity and the simplification.

a^2-b^2=(a-b)(a+b)

Transform one side at a time and make each equality readable.
Name the identity or algebra rule that justifies a significant rewrite.
Finish at the other side without changing the meaning of the expression.

4. Applying the method and checking the result

In the worked proof, the left side contains secant and tangent. Their definitions turn both into expressions involving sine and cosine. This makes it possible to combine terms over a common denominator and use the Pythagorean identity. The proof does not need a diagram or a list of angle measurements because it relies on relationships that hold throughout the expressions' domain.

After a proof, check its structure. Did you begin with one side? Is every change justified by an identity or algebra rule? Does the final expression match the other side? Are there any denominators that restrict the allowed angles? These checks help catch errors that a calculator check might miss.

A numerical check can still be a useful first step when you are unsure whether a proposed identity is plausible. Choose an angle where all terms are defined and compare both sides. If the results differ, the statement is not an identity. If they match, continue with a proof; one successful test is not enough to prove it.

Use definitions to rewrite less familiar ratios in terms of sine and cosine.
Use algebra to expose a known identity.
Numerical checks can disprove a claim but cannot prove it for every allowed angle.

Worked example

Rewrite secant and tangent using sine and cosine

Prove that

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

for every angle where the left side is defined.

Set the domain
The left side contains $\sec x$ and $\tan x$ . Both require $\cos x\ne 0$ . In addition, the full fraction requires $\tan x\ne 0$ , so $\sin x\ne 0$ . The proof will apply where both conditions hold.
$\sin x\ne 0,\quad \cos x\ne 0$
Use the reciprocal and quotient identities
Start with the left side. Replace secant with $1/\cos x$ and tangent with $\sin x/\cos x$ . These replacements follow from the definitions of the reciprocal and quotient ratios.
$\frac{\sec x-\cos x}{\tan x}=\frac{\frac{1}{\cos x}-\cos x}{\frac{\sin x}{\cos x}}$
Combine the terms in the numerator
Write the numerator over a common denominator. This turns the difference into a single fraction and makes the Pythagorean identity visible.
$\frac{\frac{1-\cos^2x}{\cos x}}{\frac{\sin x}{\cos x}}$
Apply the Pythagorean identity
Since $\sin^2x+\cos^2x=1$ , subtracting $\cos^2x$ gives $1-\cos^2x=\sin^2x$ . Substitute this equal expression into the numerator.
$\frac{\frac{\sin^2x}{\cos x}}{\frac{\sin x}{\cos x}}$
Simplify the complex fraction
Dividing by a fraction means multiplying by its reciprocal. The factors $\cos x$ and one factor of $\sin x$ cancel because both are nonzero on the stated domain. The result is the right side.
$\frac{\sin^2x}{\cos x}\cdot\frac{\cos x}{\sin x}=\sin x$

Answer: Therefore,

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

wherever the original left side is defined.

Check: The original denominator is

\tan x

, and secant and tangent also require

\cos x\ne 0

. Thus the proof's restrictions,

\sin x\ne 0

and

\cos x\ne 0

, match the domain of the original expression.

Common mistakes and how to avoid them

Checking several angles and calling the identity proved.

Correction: A few values can test a claim but cannot show it is true for every allowed angle. Use identities and algebra to create a general chain of equal expressions.

Changing both sides at once without showing how they are connected.

Correction: Work from one side toward the other. This makes the reasoning clear and helps reveal an invalid step.

Cancelling a factor that might be zero.

Correction: State the domain and confirm that the factor is nonzero before cancelling it. Do not assign a value to an expression at an angle where it is undefined.

Using an identity in the wrong direction or misremembering its sign.

Correction: Check the relationship before substituting. For example,

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

, while

1+\cos^2x

does not equal

\sin^2x

.

Lesson summary

A trigonometric identity is true for every input where both sides are defined.
The reciprocal, quotient, and Pythagorean identities provide the main substitutions used in these proofs.
Start with one side, justify each rewrite, and continue until it matches the other side.
Keep track of denominators and nonzero factors, especially when cancelling.
Numerical checks can help test a claim, but a proof must establish it generally.

Check your understanding

Question 1

Which expression is equal to

1-\sin^2x

?

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

Show answer and explanation

\cos^2x

From

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

, subtract

\sin^2x

from both sides to get

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

.

Question 2

Why is testing an identity at three angles not a proof?

An identity must be checked only in degrees.
Three successful tests do not show that equality holds for every allowed angle.
Trigonometric ratios cannot be evaluated at specific angles.
A proof can use only the Pythagorean identity.

Show answer and explanation

Three successful tests do not show that equality holds for every allowed angle.

A test checks only selected inputs. A proof uses valid relationships and algebra to establish equality for all inputs in the domain.

Question 3

For which condition is

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

defined?

$\sin x\ne 0$
$\cos x\ne 0$
$\tan x\ne 0$
$\sin x=\cos x$

Show answer and explanation

\cos x\ne 0

The quotient has

\cos x

as its denominator, so it is defined when

\cos x\ne 0

.

Key terms

Identity: An equality that is true for every allowed 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, such as $\sec x=1/\cos x$ .
Quotient identity: A relationship that expresses tangent or cotangent as a ratio of sine and cosine.
Pythagorean identity: A relationship among squared trigonometric ratios, including $\sin^2x+\cos^2x=1$ .

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation B3.3. It is a study resource, not an official curriculum publication.

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B3.3 · Prove trigonometric identities

Ontario Grade 12 Mathematics

A Grade 12 guide to using known relationships to show two expressions are equal

What you will learn

1. Prerequisite bridge: equations, ratios, and domains

2. The main relationships and how to choose one

3. A proof as a chain of equivalent expressions

4. Applying the method and checking the result

Rewrite secant and tangent using sine and cosine

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MHF4U

About this lesson