The Identity Visualizer

Simplifying with identities, Pythagorean and cofunction identities, verifying identities (9.1, 9.2, 9.3)

There is a page of identities in the back of every trig book and it looks like arbitrary trivia to memorize. It is not. Almost all of it comes from one fact you already know: the unit circle has radius 1.

Step 1 of 7: The Point on the Unit Circle

A point sits on the unit circle. You know this from your work with the unit circle already.

What is the x-coordinate of this point?

Step 2 of 7: The Circle's Equation

The point sits on a circle of radius 1 centered at the origin. You remember the equation of a circle from algebra. What is the equation of this specific circle?

Step 3 of 7: The Substitution

You know \(x = \cos\theta\) and \(y = \sin\theta\). Substitute these into \(x^2 + y^2 = 1\). What do you get?

Step 4 of 7: Two More Identities from One

Start with \(\cos^2\theta + \sin^2\theta = 1\). Divide every term by \(\cos^2\theta\). What do you get?

Step 5 of 7: Cofunction Identities

Look at this right triangle. The two acute angles are \(A\) and \(B\). Since the angles of a triangle add to 180 degrees and the right angle uses 90 of them, \(A + B = 90\) degrees, always.

The side opposite angle \(A\) is the same side that is adjacent to angle \(B\). That means \(\sin A\) and \(\cos B\) use the same two sides in the same ratio.

Since \(B = 90° - A\), what does \(\sin A\) equal?

Step 6 of 7: Identity vs. Equation

Test each statement below. The simulation will evaluate both sides at 100 random angles and graph the results. Try at least two, then answer the question.

Step 7 of 7: Guided Verification

Verify that the left side equals the right side by choosing the right algebraic move at each step. A short list of moves handles almost everything.

Target: show that \(\csc x - \sin x = \cos x \cdot \cot x\)

\(\csc x - \sin x\)

Choose your next move:

Moves: 0

Select an option in Step 1 to begin.

Identity Tester

Choose an expression for each side, then test whether they form an identity. The simulation evaluates both sides at 100 random angles and graphs the results.

Important: Numerical agreement at 100 points is strong evidence, not proof. A true proof requires algebraic verification -- showing each step follows from known identities. Testing can suggest an identity is true (or quickly show it is false), but it cannot prove one.

Left side

Right side

From the Unit Circle (Pythagorean Identities)

All three come from \(x^2 + y^2 = 1\) with \(x = \cos\theta\), \(y = \sin\theta\).

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

\(1 + \tan^2\theta = \sec^2\theta\) — divide by \(\cos^2\theta\)

\(\cot^2\theta + 1 = \csc^2\theta\) — divide by \(\sin^2\theta\)

Reciprocal Identities

\(\csc\theta = \dfrac{1}{\sin\theta}\)

\(\sec\theta = \dfrac{1}{\cos\theta}\)

\(\cot\theta = \dfrac{1}{\tan\theta}\)

Quotient Identities

\(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\)

\(\cot\theta = \dfrac{\cos\theta}{\sin\theta}\)

Cofunction Identities

From relabeling the sides of a right triangle when \(A + B = 90°\).

\(\sin\theta = \cos(90° - \theta)\)

\(\cos\theta = \sin(90° - \theta)\)

\(\tan\theta = \cot(90° - \theta)\)

Try This

Starter: Simplify an Expression

Simplify \((\sin x)(\csc x) + \cos^2 x\).

Which identity did each step use?

What does the expression simplify to?

Stretch: Spot the Key Move

Verification 1: Verify that \(\dfrac{1 - \cos^2 x}{\sin x} = \sin x\).

Which move opens up this verification?

Verification 2: Verify that \(\sec x - \cos x = \sin x \tan x\).

Which move opens up this verification?

Challenge: Fewest Moves

Verify that \(\dfrac{1 + \tan^2 x}{\csc^2 x} = \tan^2 x\).

Work from the left side. Choose the best sequence of moves.

\(\dfrac{1 + \tan^2 x}{\csc^2 x}\)

Target: \(\tan^2 x\)

Moves: 0

Now think about it from the other direction: starting from \(\tan^2 x\) and trying to reach \(\dfrac{1 + \tan^2 x}{\csc^2 x}\). Which direction was easier, and why?

Derive all three Pythagorean identities from the equation of the unit circle
Explain cofunction identities using the geometry of a right triangle
Distinguish between an identity (true for all values) and an equation (true for some values)
Verify trigonometric identities using a systematic strategy: convert to sine and cosine, apply Pythagorean identities, find common denominators, and factor
Simplify trigonometric expressions using fundamental identities

Quick Check

Which of the following is not an identity?

Instructor Notes

Teaching Notes

Build the Pythagorean identity from what students already know: the unit circle coordinates (\(\cos\theta, \sin\theta\)) and the equation of a circle (\(x^2 + y^2 = r^2\)). This makes the identity feel inevitable rather than arbitrary.
Emphasize that the three Pythagorean identities are one identity with two algebraic manipulations. Students who can re-derive them in seconds do not need to memorize them.
Cofunction identities are best taught through the right triangle, where "opposite to A is adjacent to B" is visually obvious. The word "cofunction" literally means "function of the complement."
For verification, teach the strategy hierarchy: (1) convert to sin and cos, (2) find common denominators, (3) apply Pythagorean identity, (4) factor. "Convert to sin and cos" alone handles a large fraction of textbook verification problems.

Common Student Errors

Checking a few values and concluding an identity is proven. The numerical tester in this simulation deliberately surfaces this misconception.
Working on both sides of an identity simultaneously and calling it a verification. Verification works from one side to the other, not both toward the middle.
Recognizing \(\sin^2 x + \cos^2 x = 1\) but failing to apply rearrangements like \(1 - \sin^2 x = \cos^2 x\) or \(\sin^2 x = 1 - \cos^2 x\).
Confusing identities (true for all values) with equations (true for some). The expression \(\sin x + \cos x = 1\) is a common trap.

Discussion Questions

Why do we always start verification from the more complex side? What happens if you start from the simpler side?
"Convert everything to sine and cosine" works surprisingly often. Why does reducing to two functions help so much?
How many independent identities do you actually need to memorize? If you know the Pythagorean identity and the definitions, can you re-derive everything else?
Is there a difference between "simplify" and "verify"? When does each appear on an exam?

Exam Connection

"Simplify" questions: apply one or two identities to reduce an expression to a simpler form. Often worth 3-5 points.
"Verify" questions: show a chain of equalities from one side to the other. Graded on the logical flow, not just the answer. Usually 5-8 points.
Common format: given one side, determine the other side. Students need to both simplify and recognize the result.
The Pythagorean identity and its rearrangements appear in nearly every trig exam. Students who can produce all three forms on demand have a significant advantage.