AP®MathsCollege BoardPrecalculusRevision NotesTrigonometric & Polar FunctionsTrigonometric Equations, Inequalities & IdentitiesThe Pythagorean Trigonometric Identity

The Pythagorean Trigonometric Identity (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 15 April 2026

The Pythagorean identity

What is the Pythagorean trigonometric identity?

The Pythagorean trigonometric identity is:

$\sin^{2} θ + \cos^{2} θ = 1$

This holds for every value of $θ$
- It is a true statement for all real numbers
The identity comes directly from the Pythagorean Theorem applied to the unit circle
- For any angle $θ$ in standard position, the terminal ray meets the unit circle at the point $(\cos θ, \sin θ)$
- The right triangle formed has legs of length $| \cos θ |$ and $| \sin θ |$
  - and hypotenuse $1$ (because the hypotenuse is the radius of the circle)
- Applying the Pythagorean Theorem gives $\cos^{2} θ + \sin^{2} θ = 1$

What useful rearrangements of the Pythagorean identity exist?

The identity can be rearranged in several useful ways
- These rearrangements are essential for rewriting trigonometric expressions
Simply rearranging the terms in the identity
- gives the two forms
  - $\sin^{2} θ = 1 - \cos^{2} θ$
  - $\cos^{2} θ = 1 - \sin^{2} θ$

Dividing the original identity by $\cos^{2} θ$
- gives a form involving tangent and secant
  - $\tan^{2} θ + 1 = \sec^{2} θ$
- which can also be written as
  - $\sec^{2} θ - 1 = \tan^{2} θ$

Dividing the original identity by $\sin^{2} θ$
- gives a form involving cotangent and cosecant:
  - $1 + \cot^{2} θ = \csc^{2} θ$
- which can also be written as
  - $\csc^{2} θ - 1 = \cot^{2} θ$

Examiner Tips and Tricks

As long as you remember the basic Pythagorean identity $\sin^{2} θ + \cos^{2} θ = 1$ , you can always recreate the other identities by rearranging the basic one as shown above.

How is the Pythagorean identity used to rewrite trigonometric expressions?

A common task is to rewrite a trigonometric expression as a single term involving only one specified function
When approaching a question like this
- Look for combinations like $1 - \sin^{2} x$ , $\sec^{2} x - 1$ , etc.
  - These can be replaced using one of the Pythagorean identities
- If needed, convert any remaining reciprocal trig functions ( $\sec$ , $\csc$ , $\cot$ ) into expressions involving $\sin$ and $\cos$
  - Similarly you can convert regular trig functions into reciprocal ones if necessary
- Simplify the resulting expression algebraically
  - Cancelling factors, combining fractions, etc.
These sorts of steps will let you convert the final expression back into the requested form
- e.g. an expression involving $\tan x$ only

Examiner Tips and Tricks

When an exam question asks you to rewrite a trigonometric expression, look first for patterns that match one of the Pythagorean identities.

Especially the rearranged forms like $1 - \sin^{2} x$ , $\sec^{2} x - 1$ , or $\csc^{2} x - 1$

These rearrangements appear frequently in exam questions, and recognizing them quickly is the key to making progress.

The chief reader reports consistently note that students often make the substitution but then fail to simplify the expression all the way to the requested final form

So make sure your final answer matches exactly what the question asks for (e.g. "a single term involving $\tan x$ ")

Worked Example

Rewrite the function $h (x) = \frac{1 - \cos^{2} x}{\sin x}$ as an expression in which $\sin x$ appears once and no other trigonometric functions are involved.

Answer:

The numerator $1 - \cos^{2} x$ matches the rearranged Pythagorean identity $1 - \cos^{2} x = \sin^{2} x$

Substitute this into the expression

$h (x) = \frac{\sin^{2} x}{\sin x}$

Simplify by cancelling one factor of $\sin x$

$h (x) = \sin x$

Worked Example

(a) Rewrite the function $f (x) = \frac{\sec^{2} x - 1}{1 - \cos^{2} x}$ as a single trigonometric term in which no other trigonometric functions are involved.

Answer:

Use two of the rearranged Pythagorean identities

Numerator: $\sec^{2} x - 1 = \tan^{2} x$
Denominator: $1 - \cos^{2} x = \sin^{2} x$

Substitute both into the expression

$f (x) = \frac{\tan^{2} x}{\sin^{2} x}$

Substitute in $\tan^{2} x = {(\frac{\sin x}{\cos x})}^{2} = \frac{\sin^{2} x}{\cos^{2} x}$

then simplify

$f (x) = \frac{\sin^{2} x / \cos^{2} x}{\sin^{2} x} = \frac{\sin^{2} x}{\cos^{2} x} \cdot \frac{1}{\sin^{2} x} = \frac{1}{\cos^{2} x}$

And $\frac{1}{\cos^{2} x} = {(\frac{1}{\cos x})}^{2} = {(\sec x)}^{2}$ , so

$f (x) = \sec^{2} x$

(b) Rewrite the function $g (x) = (\csc^{2} x - 1) \sin^{2} x$ as a single trigonometric term in which no other trigonometric functions are involved.

Answer:

Use the rearranged Pythagorean identity $\csc^{2} x - 1 = \cot^{2} x$

Substitute this into the expression

$g (x) = \cot^{2} x \cdot \sin^{2} x$

Substitute in $\cot^{2} x = {(\frac{\cos x}{\sin x})}^{2} = \frac{\cos^{2} x}{\sin^{2} x}$

$g (x) = \frac{\cos^{2} x}{\sin^{2} x} \cdot \sin^{2} x$

Cancel $\sin^{2} x$

$g (x) = \cos^{2} x$

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The Pythagorean Trigonometric Identity (College Board AP® Precalculus): Revision Note

The Pythagorean identity

What is the Pythagorean trigonometric identity?

What useful rearrangements of the Pythagorean identity exist?

Examiner Tips and Tricks

How is the Pythagorean identity used to rewrite trigonometric expressions?

Examiner Tips and Tricks

Worked Example

Worked Example

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Author: Roger B

Reviewer: Mark Curtis