Prove simple trigonometric identities

Ontario Grade 11 Mathematics

Trigonometric Functions

MCR3U · Strand D: Trigonometric Functions · Expectation D1.5

You have already worked with the three primary trigonometric ratios — sine, cosine, and tangent — in Grade 10. In this lesson you will discover that certain equations involving those ratios are true for every possible angle, not just one specific value. These special equations are called trigonometric identities, and learning to prove them is a core skill in MCR3U. Proving an identity means showing, through a logical sequence of algebraic steps, that both sides of an equation are always equal. This is different from solving an equation, where you find a particular angle that makes it true. By the end of this lesson you will be able to prove simple identities confidently using a clear, structured strategy.

What you will learn

Explain what a trigonometric identity is and why it differs from a regular equation.
State and apply the Pythagorean identity and the quotient identity for tangent.
Prove simple trigonometric identities by working with one side at a time using algebraic manipulation and known identities.
Identify common errors in attempted proofs and correct them.

Prerequisite Bridge: The Three Primary Ratios and the Unit Circle Connection

Recall from Grade 10 that for an angle

\theta

in a right triangle, or more generally for any angle placed in standard position, the primary trigonometric ratios are defined as

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}

,

\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}

, and

\tan\theta = \frac{\text{opposite}}{\text{adjacent}}

.

A key fact you need right now is that these three ratios are not independent — they are linked by two fundamental relationships that hold for every angle

\theta

. Those relationships are the identities you will use as building blocks throughout this lesson.

Before moving on, make sure you remember how to factor simple expressions, find a common denominator for fractions, and expand products like

(a + b)^2

. You will use each of those algebra skills when manipulating the sides of an identity.

$\sin\theta$ , $\cos\theta$ , and $\tan\theta$ are defined as ratios of side lengths (or coordinates) and are connected to each other.
All three ratios are functions of angle $\theta$ , measured here in degrees.
Algebra skills — factoring, expanding, and working with fractions — are essential tools for proving identities.

Two Fundamental Identities You Must Know

The first identity comes directly from the Pythagorean theorem. If you place an angle

\theta

in standard position and draw a point on the terminal arm at distance

r

from the origin, the coordinates of that point are

(x, y)

where

x = r\cos\theta

and

y = r\sin\theta

. Because

x^2 + y^2 = r^2

, dividing every term by

r^2

gives the Pythagorean identity.

The second identity connects tangent to the ratio of sine over cosine. Because

\tan\theta = \frac{\text{opposite}}{\text{adjacent}}

, and

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}

while

\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}

, dividing

\sin\theta

by

\cos\theta

cancels the hypotenuse and leaves

\frac{\text{opposite}}{\text{adjacent}} = \tan\theta

. This is the quotient identity.

These two identities — and rearrangements of them — are your main tools. You may rearrange the Pythagorean identity to get

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

or

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

whenever that form is more useful.

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

Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$ .
Quotient identity: $\tan\theta = \frac{\sin\theta}{\cos\theta}$ .
Useful rearrangements: $\sin^2\theta = 1 - \cos^2\theta$ and $\cos^2\theta = 1 - \sin^2\theta$ .
These identities hold for every angle $\theta$ where the ratios are defined.

What It Means to Prove an Identity

An identity is a statement that two expressions are always equal. Proving it means showing that one side can be rewritten, step by step, until it looks exactly like the other side. Think of it as simplifying or transforming one expression rather than solving for an unknown.

The most reliable strategy is to choose the more complicated-looking side and work on it alone. You apply known identities, algebra rules, or fraction operations to rewrite it. You stop when you reach an expression that is identical to the other side. You never move terms across the equals sign — that would assume the very thing you are trying to prove.

Sometimes both sides are equally complex. In that case, you may simplify each side independently until both reach the same expression. Either way, the key rule is: never treat the identity as an equation you can rearrange. Work within one side at a time.

It also helps to scan both sides before you start. Ask yourself: Does one side have fractions and the other does not? Does one side involve

\tan\theta

and the other only

\sin\theta

and

\cos\theta

? Those observations suggest which identity to use first.

Work on one side only; never add, subtract, or move terms across the equals sign.
Choose the more complex side to start.
Use the Pythagorean identity, the quotient identity, or algebra rules to rewrite expressions.
Stop when your working side matches the other side exactly.
Scanning both sides before starting helps you plan which identity to use.

Step-by-Step Strategy in Practice

Follow this four-step approach for every proof. First, write the identity with the left-hand side (LS) and right-hand side (RS) labelled separately. Second, decide which side to work on. Third, apply one algebraic or identity step at a time, writing the reason beside each line. Fourth, conclude by stating that LS = RS.

Writing a reason for each step is not just a classroom formality. It keeps your thinking clear and makes it easy to spot an error if something goes wrong. Common reasons you will write include 'quotient identity', 'Pythagorean identity', 'common denominator', 'factor', and 'cancel common factor'.

You do not need to simplify in one giant leap. Small, clear steps are always better than a rushed shortcut that is hard to verify. Each step should change the expression in exactly one way.

Label LS and RS at the top, then work on only one.
Write a brief reason beside every step.
Take small, single-change steps rather than large jumps.
Finish with a clear statement: LS = RS, therefore the identity is proven.

When Expressions Are Not Defined

An identity is only valid for angles where every ratio in it is defined. For example,

\tan\theta

is undefined when

\cos\theta = 0

, which happens at

\theta = 90°

and

\theta = 270°

. Whenever an identity involves

\tan\theta

or a denominator containing a trigonometric ratio, it is understood that the identity holds for all angles except those that make the denominator zero.

In a proof at this level, you do not need to list all excluded angles explicitly unless you are asked to do so. Simply be aware that dividing by a trigonometric expression is valid only when that expression is not zero.

$\tan\theta$ is undefined when $\cos\theta = 0$ .
Any fraction with a trigonometric denominator is undefined when that denominator equals zero.
Identities hold for all angles where every term in them is defined.

Toolkit: Identities and Algebra Moves for Proving Trigonometric Identities

Tool	Expression	When to Use It
Pythagorean identity	$\sin^2\theta + \cos^2\theta = 1$	When you see $\sin^2\theta + \cos^2\theta$ together, or need to swap $\sin^2\theta$ for $1 - \cos^2\theta$
Rearranged Pythagorean (sine form)	$\sin^2\theta = 1 - \cos^2\theta$	When the RS has $1 - \cos^2\theta$ in the numerator
Rearranged Pythagorean (cosine form)	$\cos^2\theta = 1 - \sin^2\theta$	When the RS has $1 - \sin^2\theta$ in the numerator
Quotient identity	$\tan\theta = \frac{\sin\theta}{\cos\theta}$	When $\tan\theta$ appears on one side but not the other
Common denominator	Combine fractions into one	When one side has a sum of fractions and the other has a single fraction
Cancel common factors	Divide numerator and denominator by the same expression	When the same non-zero factor appears on top and bottom

Worked example

Example 1 — Using the Quotient Identity to Simplify

Prove the identity:

\sin\theta \cdot \tan\theta = \frac{1 - \cos^2\theta}{\cos\theta}

Label and choose a side
Write the left-hand side (LS) and right-hand side (RS) separately. The LS has a product involving $\tan\theta$ , and the RS is a fraction with $\cos\theta$ in the denominator. The LS looks simpler to start with, because replacing $\tan\theta$ with a known identity will create a fraction, moving us toward the RS.
$\text{LS} = \sin\theta · \tan\theta$
Apply the quotient identity
Replace $\tan\theta$ with $\frac{\sin\theta}{\cos\theta}$ using the quotient identity. This is the key substitution that introduces $\cos\theta$ into the numerator.
$= \sin\theta · \frac{\sin\theta}{\cos\theta}$
Multiply the numerators
Multiply $\sin\theta$ by $\sin\theta$ in the numerator. Recall that $\sin\theta \cdot \sin\theta = \sin^2\theta$ .
$= \frac{\sin^2\theta}{\cos\theta}$
Apply the Pythagorean identity
Replace $\sin^2\theta$ in the numerator using the rearranged Pythagorean identity $\sin^2\theta = 1 - \cos^2\theta$ . This matches the numerator of the RS exactly.
$= \frac{1 - \cos^2\theta}{\cos\theta}$
Conclude
The expression now looks exactly like the RS, so the proof is complete.
$= \text{RS}$

Answer: LS

= \frac{1 - \cos^2\theta}{\cos\theta} =

RS, therefore the identity is proven.

Check: Verify with a specific angle. Let

\theta = 60°

. LS:

\sin 60° \cdot \tan 60° = \frac{\sqrt{3}}{2} \cdot \sqrt{3} = \frac{3}{2}

. RS:

\frac{1 - \cos^2 60°}{\cos 60°} = \frac{1 - \frac{1}{4}}{\frac{1}{2}} = \frac{\frac{3}{4}}{\frac{1}{2}} = \frac{3}{2}

. Both sides equal

\frac{3}{2}

, confirming the result.

Worked example

Example 2 — Working with a Fraction on the More Complex Side

Prove the identity:

\frac{\sin^2\theta + \cos^2\theta}{\cos\theta} = \frac{1}{\cos\theta}

Label and choose a side
Label the left-hand side (LS) and right-hand side (RS). The LS has a compound numerator, making it more complex. Work on the LS.
$\text{LS} = \frac{\sin^2\theta + \cos^2\theta}{\cos\theta}$
Apply the Pythagorean identity to the numerator
The numerator $\sin^2\theta + \cos^2\theta$ equals $1$ by the Pythagorean identity. Replace the entire numerator with $1$ .
$= \frac{1}{\cos\theta}$
Conclude
The result is exactly the RS, so the identity is proven.
$= \text{RS}$

Answer: LS

= \frac{1}{\cos\theta} =

RS, therefore the identity is proven.

Check: Verify with

\theta = 30°

. LS numerator:

\sin^2 30° + \cos^2 30° = \frac{1}{4} + \frac{3}{4} = 1

. So LS

= \frac{1}{\cos 30°} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}

. RS

= \frac{1}{\cos 30°} = \frac{2}{\sqrt{3}}

. Both sides match.

Common mistakes and how to avoid them

Moving a term across the equals sign, treating the identity like an equation to solve.

Correction: Work entirely within one side. Never add, subtract, multiply, or divide across the equals sign during a proof.

Writing

\sin^2\theta

as

\sin(\theta^2)

or

\sin 2\theta

.

Correction:

\sin^2\theta

means

(\sin\theta)^2

, the square of the sine ratio. It is not the same as

\sin(2\theta)

.

Replacing

\tan\theta

with

\frac{\cos\theta}{\sin\theta}

(flipping the quotient identity upside down).

Correction: The quotient identity is

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

, not

\frac{\cos\theta}{\sin\theta}

. The reciprocal of tangent is cotangent, which is not used in this course.

Claiming the identity is proven by checking one or two specific angles.

Correction: Checking a specific angle is a useful verification step, but it does not constitute a proof. A proof requires showing the two sides are algebraically identical for all valid angles.

Trying to simplify both sides and then adding them or multiplying them together.

Correction: Work on each side independently and stop when both reach the same expression. Do not combine the two sides arithmetically.

Lesson summary

A trigonometric identity is an equation involving trigonometric ratios that is true for every angle where the expression is defined.
The two main building-block identities are the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ and the quotient identity $\tan\theta = \frac{\sin\theta}{\cos\theta}$ .
To prove an identity, choose the more complex side, then rewrite it step by step using known identities and algebra until it matches the other side exactly.
Never move terms across the equals sign during a proof — that assumes the very thing you are trying to show.
Write a brief reason beside every step to keep your argument clear and easy to check.
You can verify (but not prove) an identity by substituting a specific angle and confirming both sides give the same value.

Check your understanding

Question 1

Which of the following correctly states the quotient identity?

$\tan\theta = \frac{\cos\theta}{\sin\theta}$
$\tan\theta = \sin\theta \cdot \cos\theta$
$\tan\theta = \frac{\sin\theta}{\cos\theta}$
$\tan\theta = \frac{1}{\sin\theta}$

Show answer and explanation

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

The quotient identity states

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

. Option A is the reciprocal of tangent (cotangent). Option B is a product, not a quotient. Option D is the reciprocal of sine (cosecant).

Question 2

You are proving an identity and the left-hand side currently reads

\frac{\sin^2\theta}{\cos\theta}

. You want to rewrite the numerator so it matches the right-hand side, which is

\frac{1 - \cos^2\theta}{\cos\theta}

. Which identity lets you do that?

The quotient identity: $\tan\theta = \frac{\sin\theta}{\cos\theta}$
The Pythagorean identity rearranged: $\sin^2\theta = 1 - \cos^2\theta$
The Pythagorean identity rearranged: $\cos^2\theta = 1 - \sin^2\theta$
No identity is needed; the two expressions are already different.

Show answer and explanation

The Pythagorean identity rearranged:

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

Replacing

\sin^2\theta

with

1 - \cos^2\theta

uses the rearranged Pythagorean identity

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

. This transforms the numerator so it matches the RS exactly.

Question 3

A student proves an identity by substituting

\theta = 45°

and showing both sides equal the same number. Is this a valid proof?

Yes, because one example is enough to confirm an identity.
Yes, because CAD 45° is the most common angle used in proofs.
No, because you also need to check $\theta = 90°$ .
No, because a proof must show both sides are algebraically identical for all valid angles, not just one.

Show answer and explanation

No, because a proof must show both sides are algebraically identical for all valid angles, not just one.

Checking a specific angle is only a verification step. A true proof uses algebra and known identities to show the two sides are equal for every valid angle, not just one particular value.

Question 4

To prove the identity

\cos\theta \cdot \tan\theta = \sin\theta

, a student works on the left-hand side and replaces

\tan\theta

with

\frac{\sin\theta}{\cos\theta}

. What is the correct next step?

Replace $\cos\theta$ with $1 - \sin^2\theta$ .
Multiply $\cos\theta$ by $\frac{\sin\theta}{\cos\theta}$ and cancel $\cos\theta$ from the numerator and denominator.
Move $\sin\theta$ to the right-hand side.
Square both sides to remove the trigonometric ratios.

Show answer and explanation

Multiply

\cos\theta

by

\frac{\sin\theta}{\cos\theta}

and cancel

\cos\theta

from the numerator and denominator.

After substituting the quotient identity, the LS becomes

\cos\theta \cdot \frac{\sin\theta}{\cos\theta}

. Multiplying and cancelling the common factor

\cos\theta

gives

\sin\theta

, which equals the RS. Moving terms across or squaring both sides are not permitted strategies in an identity proof.

Key terms

Trigonometric identity: An equation involving trigonometric ratios that is true for every angle for which all terms are defined, not just for specific angle values.
Pythagorean identity: The relationship $\sin^2\theta + \cos^2\theta = 1$ , which holds for all angles $\theta$ and comes directly from the Pythagorean theorem.
Quotient identity: The relationship $\tan\theta = \frac{\sin\theta}{\cos\theta}$ , which expresses tangent as the ratio of sine to cosine.
$\sin^2\theta$: Shorthand for $(\sin\theta)^2$ , meaning the sine ratio multiplied by itself. It does not mean $\sin(\theta^2)$ or $\sin(2\theta)$ .
Left-hand side (LS) / Right-hand side (RS): The expressions on either side of the equals sign in an identity. In a proof, you label them separately and work on only one at a time.
Verification: Checking that an identity holds for a specific angle value. Useful for catching errors, but not a substitute for a full algebraic proof.
Undefined expression: A trigonometric ratio that has no value for a particular angle, such as $\tan 90°$ where the denominator $\cos 90° = 0$ . Identities are only valid where all their terms are defined.

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