Back to Lesson

Teacher Guide: Proving Trigonometric Identities

Learn how to prove a trig identity step by step: work on one side, convert to sine and cosine, combine fractions, use the conjugate, and disprove false claims with one angle.

Present LessonShare to ClassroomView Student Page

Use this lesson with your class

Free, no student accounts needed.

Share with students

Students open the lesson and practise with instant feedback.

Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Trigonometric Identities. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Explain the difference between an identity and an equation
  • Prove a trigonometric identity by transforming one side into the other
  • Choose a strategy: convert to sine and cosine, combine fractions, factor or multiply by the conjugate
  • Disprove a claimed identity with a single counterexample angle
  • Check an identity numerically and graphically and explain why this is not a proof
Prerequisites
  • • Reciprocal identities (sec⁡\sec, csc⁡\csc, cot⁡\cot) and quotient identities
  • • Pythagorean identities, especially sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1
  • • Simplifying trigonometric expressions
  • • Adding fractions with a common denominator and the difference of squares (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2
Discussion Starters
  • 1. Why is one counterexample enough to disprove a claim, but a thousand examples are not enough to prove it?
  • 2. Why are we not allowed to cross-multiply when proving an identity?
  • 3. How do you decide which side of an identity to start from?
  • 4. Where else in math have you multiplied by a form of 1, like 1+cos⁡x1+cos⁡x\frac{1 + \cos x}{1 + \cos x}?
Common Misconceptions

A proof is finished as soon as both sides give the same number at one angle

Remediation: Show sin⁡x+cos⁡x=1\sin x + \cos x = 1: true at 0°0° and 90°90°, false at 45°45°. Then graph both sides to see that they only meet at some points.

Proving an identity works like solving an equation (move terms, cross-multiply)

Remediation: Write the proof as a vertical chain: left side == ... == ... == right side. Every line must be equal to the line above it on its own.

Only one correct proof exists

Remediation: Let groups prove the same identity in different ways (start from the left vs from the right, conjugate vs common denominator) and compare.

Differentiation Ideas

For Struggling Students:

  • • Provide a card with all reciprocal, quotient and Pythagorean identities
  • • Start with one-step proofs such as tan⁡xcos⁡x=sin⁡x\tan x \cos x = \sin x
  • • Let students pick the next step from choices before writing proofs alone

For On-Level Students:

  • • Prove identities that need a common denominator and a Pythagorean identity
  • • Use the conjugate on fractions like sin⁡x1−cos⁡x\frac{\sin x}{1 - \cos x}
  • • Disprove false claims with a counterexample angle

For Advanced Students:

  • • Prove identities that need several strategies, like cos⁡x1−sin⁡x=sec⁡x+tan⁡x\frac{\cos x}{1 - \sin x} = \sec x + \tan x
  • • Find all angles where a proved identity is not defined
  • • Write their own false identity that holds at three angles and challenge classmates to disprove it
Standards Alignment
  • HSF-TF.C.8 (CCSS.MATH.CONTENT.HSF.TF.C.8)

    Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ)

  • HSA-SSE.A.2 (CCSS.MATH.CONTENT.HSA.SSE.A.2)

    Use the structure of an expression to identify ways to rewrite it

Lesson Resources
  • visualIdentity Proof Stepper

    Students choose the next step of a proof and see the rule behind it, with a live numeric check of both sides

  • activityIdentity or Not?

    Pairs graph both sides of several claims and look for a counterexample angle before trying a proof

  • worksheetProof Strategies

    Ten identities sorted by strategy: convert, combine, factor, conjugate

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A trigonometric identity is an equation that is true for every angle xx where both sides are defined. An ordinary equation like sin⁡x=12\sin x = \frac{1}{2} is true only for some angles, and you *solve* it. An identity like tan⁡xcos⁡x=sin⁡x\tan x \cos x = \sin x is true for all allowed angles, and you *prove* it.

What proving means

To prove an identity, you start from one side and change it, one legal step at a time, until it looks exactly like the other side:
tan⁡xcos⁡x⏟left side=sin⁡xcos⁡x⋅cos⁡x=sin⁡x⏟right side\underbrace{\tan x \cos x}_{\text{left side}} = \frac{\sin x}{\cos x}\cdot\cos x = \underbrace{\sin x}_{\text{right side}}

Rules of the game

  • Work on one side only (usually the more complicated one). The other side stays as it is.
  • Never move terms across the equals sign and never multiply or divide both sides: that would assume the identity is already true.
  • Every step must use an identity you know (reciprocal, quotient, Pythagorean) or correct algebra (common denominators, factoring, cancelling).

Strategy toolbox

  1. 1.Start from the more complicated side.
  2. 2.Convert to sin⁡\sin and cos⁡\cos (tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x}, sec⁡x=1cos⁡x\sec x = \frac{1}{\cos x}, ...).
  3. 3.Combine fractions over a common denominator.
  4. 4.Factor (common factors, difference of squares).
  5. 5.Multiply by the conjugate, e.g. 1+cos⁡x1+cos⁡x\frac{1 + \cos x}{1 + \cos x}, to create 1−cos⁡2x=sin⁡2x1 - \cos^2 x = \sin^2 x.
  6. 6.Use the Pythagorean identities sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1, 1+tan⁡2x=sec⁡2x1 + \tan^2 x = \sec^2 x, 1+cot⁡2x=csc⁡2x1 + \cot^2 x = \csc^2 x.

Disproving

One angle where the two sides give different values (a counterexample) is enough to show that a claim is not an identity. Checking a few angles numerically is a good test, but it never replaces a proof.

Worked Examples

Prove that tan⁡xcos⁡x=sin⁡x\tan x \cos x = \sin x.
1

Choose the side to work on

The left side has two functions, the right side only one. Start from the left. → tan⁡xcos⁡x\tan x \cos x
2

Use the quotient identity

Replace tan⁡x\tan x with sin⁡xcos⁡x\frac{\sin x}{\cos x} → sin⁡xcos⁡x⋅cos⁡x\frac{\sin x}{\cos x} \cdot \cos x
3

Cancel the common factor

cos⁡x\cos x in the numerator and denominator cancel (where cos⁡x≠0\cos x \neq 0) → sin⁡x\sin x
4

Compare with the right side

The left side now looks exactly like the right side → sin⁡x=sin⁡x\sin x = \sin x ✓

Common Mistakes

Working on both sides like an equation, for example cross-multiplying sin⁡x1−cos⁡x=1+cos⁡xsin⁡x\frac{\sin x}{1 - \cos x} = \frac{1 + \cos x}{\sin x} to get sin⁡2x=1−cos⁡2x\sin^2 x = 1 - \cos^2 x

Why it's wrong: Cross-multiplying, moving terms or dividing both sides all start from the claim that the two sides are equal, which is exactly what you are trying to show.

Correct: Transform one side until it becomes the other side. The equals sign stays untouched until the very end.

Checking one or two angles and calling the identity proved

Why it's wrong: sin⁡x+cos⁡x=1\sin x + \cos x = 1 works at 0°0° and 90°90° but fails at 45°45°. A few true examples never prove that something holds for every angle.

Correct: Use numbers only to test or to disprove. A proof needs algebra with identities that hold for all xx.

Splitting a fraction at the denominator: sin⁡x1−cos⁡x=sin⁡x1−sin⁡xcos⁡x\frac{\sin x}{1 - \cos x} = \frac{\sin x}{1} - \frac{\sin x}{\cos x}

Why it's wrong: You may split a sum in the numerator, a+bc=ac+bc\frac{a + b}{c} = \frac{a}{c} + \frac{b}{c}, but never a sum in the denominator. At x=60°x = 60° the left side is 1.7321.732 and the right side is −0.866-0.866.

Correct: To get rid of 1−cos⁡x1 - \cos x in a denominator, multiply numerator and denominator by the conjugate 1+cos⁡x1 + \cos x.

Writing (1−cos⁡x)(1+cos⁡x)=1−cos⁡x(1 - \cos x)(1 + \cos x) = 1 - \cos x or 1+cos⁡2x1 + \cos^2 x

Why it's wrong: The product of conjugates is a difference of squares: both terms are squared and the sign is minus.

Correct: (1−cos⁡x)(1+cos⁡x)=1−cos⁡2x=sin⁡2x(1 - \cos x)(1 + \cos x) = 1 - \cos^2 x = \sin^2 x

Why It Matters

Proving identities trains the most useful skill in algebra: rewriting an expression into a smarter form without changing its value.
  • Calculus: integrals like ∫sin⁡2x dx\int \sin^2 x\,dx or ∫sec⁡x dx\int \sec x\,dx only become solvable after an identity rewrites them.
  • Physics and engineering: wave and signal formulas are simplified with identities before anything is computed.
  • Computer graphics: rotations keep lengths because cos⁡2θ+sin⁡2θ=1\cos^2\theta + \sin^2\theta = 1.
  • Logical thinking: a proof is a chain of justified steps, the same kind of reasoning used in programming and in every later math course.
  • Checking answers: knowing that two different looking answers are the same identity saves you from thinking one of them is wrong.

Real World Applications

How Computer Algebra Tests Identities

Math software checks whether two formulas are the same by plugging in many random angles before it tries a full symbolic proof. If even one angle gives different values, the formulas are different.

Example:

To compare sec⁡x−cos⁡x\sec x - \cos x and sin⁡xtan⁡x\sin x \tan x, a program evaluates both at angles like 17°17°, 60°60° and 133°133° and finds the same values every time.

1Try It Yourself

A program tests the identity sec⁡x−cos⁡x=sin⁡xtan⁡x\sec x - \cos x = \sin x \tan x at x=60°x = 60°, where cos⁡60°=12\cos 60° = \frac{1}{2}, sec⁡60°=2\sec 60° = 2, sin⁡60°=32\sin 60° = \frac{\sqrt{3}}{2} and tan⁡60°=3\tan 60° = \sqrt{3}.

What value do both sides give at x=60°x = 60°?

Step 1: Write the mathematical expression

Compute the left side sec⁡60°−cos⁡60°\sec 60° - \cos 60°:

Physics: Simplifying a Ramp Formula

Physics formulas often come out of a calculation in a clumsy form. An identity turns them into a shorter form that is faster to compute and easier to understand.

Example:

A force on a ramp came out as F=Wsin⁡θtan⁡θF = \dfrac{W\sin\theta}{\tan\theta}. Since sin⁡θtan⁡θ=sin⁡θ⋅cos⁡θsin⁡θ=cos⁡θ\frac{\sin\theta}{\tan\theta} = \sin\theta \cdot \frac{\cos\theta}{\sin\theta} = \cos\theta, the formula is simply F=Wcos⁡θF = W\cos\theta.

2Try It Yourself

A box with weight W=200W = 200 N rests on a ramp with angle θ=60°\theta = 60°. The force is F=Wsin⁡θtan⁡θ=Wcos⁡θF = \dfrac{W\sin\theta}{\tan\theta} = W\cos\theta.

How large is the force FF in newtons?

Step 1: Write the mathematical expression

Use the simplified formula F=Wcos⁡θF = W\cos\theta:

Computer Graphics: Rotations Keep Lengths

Games rotate points with $x' = x\cos\theta - y\sin\theta$ and $y' = x\sin\theta + y\cos\theta$. Expanding $x'^2 + y'^2$ and using $\sin^2\theta + \cos^2\theta = 1$ proves that $x'^2 + y'^2 = x^2 + y^2$ for every angle, so objects never stretch.

Example:

The point (3,4)(3, 4) rotated by 30°30°, 90°90° or 217°217° always stays at distance 5 from the origin.

3Try It Yourself

A game rotates the point (3,4)(3, 4) around the origin by some angle θ\theta. The identity x′2+y′2=x2+y2x'^2 + y'^2 = x^2 + y^2 holds for every θ\theta.

How far from the origin is the rotated point?

Step 1: Write the mathematical expression

Use x2+y2\sqrt{x^2 + y^2} with the original point:

Key Takeaways

  • 1An identity is true for every allowed angle; an equation is true only for some angles
  • 2Prove an identity by transforming one side, step by step, until it matches the other side
  • 3Never move terms across the equals sign and never do the same operation to both sides
  • 4Start from the more complicated side and convert everything to sin⁡\sin and cos⁡\cos
  • 5Combine fractions, factor, multiply by the conjugate and use sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1
  • 6One counterexample angle disproves a claim; numeric checks test an identity but do not prove it

Frequently Asked Questions

Why can't I just do the same thing to both sides?

Doing the same operation to both sides starts from the statement that the sides are equal, which is what you want to prove. Some of those operations (squaring, multiplying by an expression that can be 0) can even turn a false claim into a true one. Working on one side avoids this.

Which side should I start with?

Usually the more complicated one: the side with more terms, fractions, or functions like sec⁡\sec, csc⁡\csc, tan⁡\tan, cot⁡\cot. Simplifying is easier than inventing complications.

What if I get stuck halfway?

Try another tool: convert to sin⁡\sin and cos⁡\cos, combine fractions, factor, or multiply by a conjugate. You may also simplify both sides separately until they reach the same expression, as long as each side is transformed on its own.

Is checking the graph a proof?

No. If the graphs of both sides overlap everywhere, the identity is very likely true, and if they do not overlap, it is false. But a picture only shows a limited window, so a proof still needs algebraic steps.

Glossary

Identity
An equation that is true for every value of the variable where both sides are defined, e.g. sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1
Left side / right side
The expressions to the left and to the right of the equals sign
Conjugate
For a−ba - b the conjugate is a+ba + b (and the other way round); their product is a2−b2a^2 - b^2
Common denominator
A denominator shared by several fractions, so that they can be combined into one fraction
Counterexample
One value for which a claimed identity is false; it is enough to disprove the claim

Formula Card

Reciprocal identities

sec⁡x=1cos⁡x, csc⁡x=1sin⁡x, cot⁡x=1tan⁡x\sec x = \frac{1}{\cos x},\ \csc x = \frac{1}{\sin x},\ \cot x = \frac{1}{\tan x}

Use them to convert sec, csc and cot into sin and cos

Quotient identities

tan⁡x=sin⁡xcos⁡x, cot⁡x=cos⁡xsin⁡x\tan x = \frac{\sin x}{\cos x},\ \cot x = \frac{\cos x}{\sin x}

Turn tangent and cotangent into fractions of sine and cosine

Pythagorean identities

sin⁡2x+cos⁡2x=1, 1+tan⁡2x=sec⁡2x, 1+cot⁡2x=csc⁡2x\sin^2 x + \cos^2 x = 1,\ 1 + \tan^2 x = \sec^2 x,\ 1 + \cot^2 x = \csc^2 x

Replace a sum or difference of squares by a single term

Conjugate products

(1−cos⁡x)(1+cos⁡x)=sin⁡2x, (1−sin⁡x)(1+sin⁡x)=cos⁡2x(1 - \cos x)(1 + \cos x) = \sin^2 x,\ (1 - \sin x)(1 + \sin x) = \cos^2 x

The reason why multiplying by the conjugate works

More in This Topic

Reciprocal Identities

Advanced25 min

Pythagorean Identities

Advanced25 min

Even and Odd Trigonometric Identities

Advanced25 min

Cofunction Identities

Advanced25 min