Pythagorean Trigonometric Identities Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Pythagorean Trigonometric Identities.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The fundamental identity $\sin^2\theta + \cos^2\theta = 1$ and its derived forms: $1 + \tan^2\theta = \sec^2\theta$ and $1 + \cot^2\theta = \csc^2\theta$ .

On the unit circle, the point $(\cos\theta, \sin\theta)$ is always at distance 1 from the origin. By the Pythagorean theorem, $x^2 + y^2 = 1$ becomes $\cos^2\theta + \sin^2\theta = 1$ . This single fact—that sine and cosine are tied to a circle—generates all three Pythagorean identities. Dividing through by $\cos^2\theta$ or $\sin^2\theta$ produces the other two forms.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: $\sin^2\theta+\cos^2\theta=1$ and its two divided-down forms let you swap one trig function for another.

Common stuck point: The procedure for pythagorean trigonometric identities is the easy part; the trap is reading $\sin^2\theta$ as $\sin(\theta^2)$ . Asking "Do I have squared trig functions I can collapse to 1 or swap using the unit-circle relation?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do I have squared trig functions I can collapse to 1 or swap using the unit-circle relation?

Worked Examples

Example 1

easy

\sin(\theta) = \frac{3}{5}

and

\theta

is in Quadrant I, find

\cos(\theta)

using the Pythagorean identity.

Answer

\cos (θ) = \frac{4}{5}

First step

Start with the Pythagorean identity:

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

Full solution

2
Substitute $\sin(\theta) = \frac{3}{5}$ : $\left(\frac{3}{5}\right)^2 + \cos^2(\theta) = 1$ , so $\frac{9}{25} + \cos^2(\theta) = 1$ .
3
Solve: $\cos^2(\theta) = 1 - \frac{9}{25} = \frac{16}{25}$ , so $\cos(\theta) = \pm\frac{4}{5}$ .
4
Since $\theta$ is in Quadrant I, $\cos(\theta) > 0$ , so $\cos(\theta) = \frac{4}{5}$ .

The Pythagorean identity

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

directly relates sine and cosine. When you know one and the quadrant, you can find the other. The quadrant determines the sign of the result.

Example 2

medium

Simplify the expression

\frac{1 - \cos^2(\theta)}{\sin(\theta) \cos(\theta)}

Example 3

medium

Verify

\sin x(\csc x - \sin x) = \cos^2 x

Example 4

hard

Verify the identity

\dfrac{1 + \sin\theta}{\cos\theta} + \dfrac{\cos\theta}{1 + \sin\theta} = 2\sec\theta

Example 5

hard

Verify

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

Example 6

challenge

\sin\theta + \cos\theta = \tfrac{\sqrt{2}}{2}

, find

\sin^3\theta + \cos^3\theta

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

Prove that

\tan^2(\theta) + 1 = \sec^2(\theta)

Example 2

hard

Simplify

\frac{\sec^2(\theta) - 1}{\csc^2(\theta) - 1}

Example 3

easy

State the fundamental Pythagorean identity.

Example 4

easy

\sin\theta = \frac{3}{5}

, find

\cos^2\theta

Example 5

easy

Simplify

1 - \sin^2\theta

Example 6

easy

State the identity relating

\tan\theta

and

\sec\theta

Example 7

easy

Simplify

\sin^2\theta + \cos^2\theta + 3

Example 8

easy

Simplify

1 - \cos^2\theta

Example 9

easy

State the identity relating

\cot\theta

and

\csc\theta

Example 10

easy

Simplify

\sec^2\theta - \tan^2\theta

Example 11

medium

\cos\theta = -\frac{4}{5}

and

\theta

is in Quadrant II, find

\sin\theta

Example 12

medium

\tan\theta = \frac{5}{12}

and

\theta

is in Quadrant I, find

\sec\theta

Example 13

medium

Simplify

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

(assume

\sin\theta \neq 0

Example 14

medium

Verify the identity

\frac{\cos\theta}{1 - \sin\theta} = \frac{1 + \sin\theta}{\cos\theta}

by cross-multiplying.

Example 15

medium

Simplify

\tan\theta \cdot \cos\theta

Example 16

medium

\sin\theta = \frac{2}{3}

and

\theta

is in Quadrant II, find

\tan\theta

Example 17

medium

Simplify

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

(assume

\cos\theta \neq -1

Example 18

medium

Simplify

\cos\theta \cdot \csc\theta \cdot \tan\theta

Example 19

medium

\sin\theta = \frac{7}{25}

and

\theta

is in Quadrant I, find

\cos\theta

Example 20

challenge

Prove the identity

\frac{1}{1 - \sin\theta} + \frac{1}{1 + \sin\theta} = 2\sec^2\theta

Example 21

challenge

\sec\theta - \tan\theta = \frac{1}{3}

, find

\sec\theta + \tan\theta

Example 22

challenge

Express

\sin^4\theta + \cos^4\theta

in terms of

\sin^2\theta\cos^2\theta

, then simplify.

Example 23

easy

Simplify

\csc^2\theta - \cot^2\theta

Example 24

easy

\cos\theta = \tfrac{1}{2}

, find

\sin^2\theta

Example 25

easy

Simplify

5(\sin^2 x + \cos^2 x)

Example 26

easy

Simplify

\csc^2 x - 1

Example 27

easy

\sin\theta = 0

, what is

\cos^2\theta

Example 28

medium

Simplify

\tan x\sin x + \cos x

Example 29

medium

\sin\theta = -\tfrac{8}{17}

and

\theta

is in Quadrant III, find

\cos\theta

Example 30

medium

\sec\theta = \tfrac{5}{3}

and

\theta

is in Quadrant IV, find

\tan\theta

Example 31

medium

Simplify

\dfrac{\cos^2 x - 1}{\cos x - 1}

Example 32

medium

Simplify

\dfrac{\tan x}{\sec x}

Example 33

medium

\tan\theta = -\tfrac{3}{4}

and

\theta

is in Quadrant II, find

\sec\theta

and

\sin\theta

Example 34

medium

Simplify

\sin^2 x\sec^2 x + \sin^2 x

Example 35

medium

Simplify

\sin x\cot x

Example 36

hard

Solve

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

[0, 2\pi)

Example 37

hard

\sin\theta + \csc\theta = 3

, find

\sin^2\theta + \csc^2\theta

Example 38

hard

Simplify

(\sec x - \tan x)(\sec x + \tan x)

Example 39

hard

Simplify

\sin^6 x + \cos^6 x

in terms of

\sin^2 x\cos^2 x

Example 40

challenge

Find all

\theta \in [0, 2\pi)

satisfying

\sec^2\theta + \tan^2\theta = 3

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Related Concepts

Trigonometric Functions Pythagorean Theorem Unit Circle Sum and Difference Identities Simplifying Rational Expressions

Background Knowledge

These ideas may be useful before you work through the harder examples.

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