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.

Advanced30 minLesson
Printable worksheet

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.

Try it now

Which statement is an identity (true for every allowed angle xx)?

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

Interactive Visual

Prove

(1+tan⁡2x)cos⁡2x=1(1 + \tan^2 x)\cos^2 x = 1

  1. Start with the left side

    (1+tan⁡2x)cos⁡2x(1 + \tan^2 x)\cos^2 x

Which expression comes next?

Check with numbers

Left side

(1+tan⁡2x)cos⁡2x(1 + \tan^2 x)\cos^2 x

= 1

Right side

11

= 1

Both sides have the same value.

Function graph

-300-200-1000100200300-2-1012xy(45, 1.41)

Tap or hover over the graph to read off values.

  • (sin a + cos a)² = 2
  • 1 + 2 sin a cos a = 2
  • sin a + cos a (should be 1?) = 1.41

Tap a name to show or hide its graph.

Interactive Sandbox

Interactive Grapher

-300-200-1000100200300-3-2-10123xy

Tap or hover over the graph to read off values.

Try these examples:

y = 2x + 1

m=2, b=1

Expression Calculator

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which statement is an identity (true for every allowed angle xx)?

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

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.
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.
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.
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.
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

Today's challengeOne problem, three tries

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