Fundamental Trig Identities (With Examples)

The fundamental trig identities are equations that are true for every angle where both sides are defined. They include the reciprocal identities, such as $\csc \theta = \frac{1}{\sin \theta}$ , the quotient identities, such as $\tan \theta = \frac{\sin \theta}{\cos \theta}$ , and the Pythagorean identities, such as $\sin^2 \theta + \cos^2 \theta = 1$ . They let you rewrite and simplify trig expressions.

Before this lesson: You should know The Unit Circle, Six Trigonometric Functions and Simplifying Rational Expressions.

Where the Identities Come From

An identity is an equation that is true for every allowed value of the variable, like $2(x + 1) = 2x + 2$ . Trig identities work the same way, with angles.

The most important one comes straight from the unit circle. The point for angle $\theta$ is $(\cos \theta, \sin \theta)$ , and every point on the circle satisfies $x^2 + y^2 = 1$ . So $\cos^2 \theta + \sin^2 \theta = 1$ for every angle. It is the Pythagorean theorem for a triangle with hypotenuse 1.

A unit circle with a point in Quadrant I at angle theta. A right triangle is drawn from the origin to the point and down to the x-axis. The horizontal leg is labeled cos theta in blue, the vertical leg is labeled sin theta in orange, and the hypotenuse, a radius, is labeled 1. — The legs of the triangle are cos θ and sin θ, and the hypotenuse is 1. By the Pythagorean theorem, sin²θ + cos²θ = 1.

The Fundamental Identities

Type	Identities
Reciprocal	$\csc \theta = \frac{1}{\sin \theta}$ , $\sec \theta = \frac{1}{\cos \theta}$ , $\cot \theta = \frac{1}{\tan \theta}$
Quotient	$\tan \theta = \frac{\sin \theta}{\cos \theta}$ , $\cot \theta = \frac{\cos \theta}{\sin \theta}$
Pythagorean	$\sin^2 \theta + \cos^2 \theta = 1$ , $1 + \tan^2 \theta = \sec^2 \theta$ , $1 + \cot^2 \theta = \csc^2 \theta$
Even-odd	$\sin(-\theta) = -\sin \theta$ , $\cos(-\theta) = \cos \theta$ , $\tan(-\theta) = -\tan \theta$
Cofunction	$\sin(\frac{\pi}{2} - \theta) = \cos \theta$ , $\cos(\frac{\pi}{2} - \theta) = \sin \theta$

The other two Pythagorean identities come from the first. Divide every term of $\sin^2 \theta + \cos^2 \theta = 1$ by $\cos^2 \theta$ to get $\tan^2 \theta + 1 = \sec^2 \theta$ . Divide by $\sin^2 \theta$ instead to get $1 + \cot^2 \theta = \csc^2 \theta$ .

Rewrite everything in terms of sine and cosine.
Look for a Pythagorean pattern, such as $1 - \cos^2 \theta$ , and replace it.
Combine fractions or cancel common factors.
Rewrite the result with the simplest function you can.

Worked Examples

Example 1: Find a value with a Pythagorean identity

If $\sin \theta = \frac{3}{5}$ and $\theta$ is in Quadrant II, find $\cos \theta$ and $\tan \theta$ .

(\frac{3}{5})^2 + \cos^2 \theta = 1

Substitute into $\sin^2 \theta + \cos^2 \theta = 1$ .

\frac{9}{25} + \cos^2 \theta = 1

Square.

\cos^2 \theta = \frac{16}{25}

Subtract $\frac{9}{25}$ from both sides.

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

Take the square root.

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

Cosine is negative in Quadrant II.

\tan \theta = \frac{\frac{3}{5}}{-\frac{4}{5}} = -\frac{3}{4}

Use the quotient identity.

Answer: $\cos \theta = -\frac{4}{5}$ and $\tan \theta = -\frac{3}{4}$

Example 2: Simplify a product

Simplify $\sin \theta \cot \theta$ .

\sin \theta \cdot \frac{\cos \theta}{\sin \theta}

Rewrite cotangent with the quotient identity.

\cos \theta

Cancel the common factor $\sin \theta$ .

Answer: $\sin \theta \cot \theta = \cos \theta$

Example 3: Use a Pythagorean identity

Simplify $\frac{1 - \cos^2 \theta}{\sin \theta}$ .

Rearrange $\sin^2 \theta + \cos^2 \theta = 1$ to get $1 - \cos^2 \theta = \sin^2 \theta$

\frac{\sin^2 \theta}{\sin \theta}

Replace the numerator.

\sin \theta

Cancel one factor of $\sin \theta$ .

Answer: $\frac{1 - \cos^2 \theta}{\sin \theta} = \sin \theta$

Example 4: Combine fractions

Simplify $\sec \theta - \sin \theta \tan \theta$ .

\frac{1}{\cos \theta} - \sin \theta \cdot \frac{\sin \theta}{\cos \theta}

Rewrite in sine and cosine.

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

Multiply.

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

Subtract the fractions, which share a denominator (see adding and subtracting rational expressions)

\frac{\cos^2 \theta}{\cos \theta}

Use the Pythagorean identity.

\cos \theta

Cancel one factor of $\cos \theta$ .

Answer: $\sec \theta - \sin \theta \tan \theta = \cos \theta$

Example 5: Word problem with a ramp

A ramp rises 1 foot for every 12 feet of horizontal run, so $\tan \theta = \frac{1}{12}$ . The secant, $\sec \theta$ , is the ramp length for each foot of run. Use an identity to find $\sec \theta$ , then find the ramp length for a 24-foot run.

\sec^2 \theta = 1 + (\frac{1}{12})^2

Use $1 + \tan^2 \theta = \sec^2 \theta$ .

\sec^2 \theta = 1 + \frac{1}{144}

Square.

\sec^2 \theta = \frac{145}{144}

Add.

\sec \theta = \frac{\sqrt{145}}{12} \approx 1.0035

Take the square root (positive for an acute angle)

24 \cdot \frac{\sqrt{145}}{12} = 2\sqrt{145} \approx 24.08

Multiply by the run.

Answer: $\sec \theta = \frac{\sqrt{145}}{12}$ , and a 24-foot run needs a ramp about 24.08 feet long.

Common Mistakes

Mistake	Example of the error	Fix
Thinking sin²θ means sin of θ²	$\sin^2 \theta = \sin(\theta^2)$	$\sin^2 \theta$ means $(\sin \theta)^2$ .
Writing the Pythagorean identity without squares	$\sin \theta + \cos \theta = 1$	The squares matter: $\sin^2 \theta + \cos^2 \theta = 1$ . For 45°, $\sin \theta + \cos \theta = \sqrt{2}$ .
Wrong sign after a square root	$\cos \theta = \frac{4}{5}$ in Quadrant II	Use the quadrant: cosine is negative in Quadrant II.
Mixing up the second identity	$1 + \tan^2 \theta = \csc^2 \theta$	Tangent pairs with secant, cotangent with cosecant: $1 + \tan^2 \theta = \sec^2 \theta$ .
Canceling terms instead of factors	$\frac{1 - \cos^2 \theta}{\cos \theta} = 1 - \cos \theta$	Only common factors cancel. Use $1 - \cos^2 \theta = \sin^2 \theta$ first.

Related Methods

Check an identity with numbers

Plug in an angle to test a simplification. For Example 4 with $\theta = 60^\circ$ : $\sec 60^\circ - \sin 60^\circ \tan 60^\circ = 2 - \frac{\sqrt{3}}{2} \cdot \sqrt{3} = 2 - \frac{3}{2} = \frac{1}{2}$ , which equals $\cos 60^\circ$ . A check like this can catch mistakes, but it does not prove the identity.

Rearranged forms

Each Pythagorean identity can be solved for any term: $\cos^2 \theta = 1 - \sin^2 \theta$ , $\tan^2 \theta = \sec^2 \theta - 1$ , $\cot^2 \theta = \csc^2 \theta - 1$ . Spotting these forms is the key skill for the next lesson.

Factoring

Expressions like $\sin^2 \theta - \cos^2 \theta$ or $\sin^2 \theta + 2 \sin \theta + 1$ factor just like polynomials, as in factoring the difference of squares.

Practice Problems

If $\cos \theta = \frac{5}{13}$ and $\theta$ is in Quadrant I, find $\sin \theta$ .
If $\sin \theta = -\frac{1}{2}$ and $\theta$ is in Quadrant III, find $\cos \theta$ .
If $\tan \theta = 2$ and $\theta$ is in Quadrant I, find $\sec \theta$ .
If $\cot \theta = \frac{3}{4}$ and $\theta$ is in Quadrant I, find $\csc \theta$ .
Simplify $\cos \theta \tan \theta$ .
Simplify $\sin^2 \theta + \cos^2 \theta + \tan^2 \theta$ .
Simplify $\frac{\sec^2 \theta - 1}{\tan \theta}$ .
Simplify $\cot \theta \sec \theta$ .
Simplify $(1 - \sin^2 \theta) \sec \theta$ .
A roof rafter has $\sec \theta = 1.25$ , where $\theta$ is the roof’s angle. Use an identity to find $\tan \theta$ , the rise per foot of run, and the rise over a 12-foot run.

Show answers

\sin^2 \theta = 1 - \frac{25}{169} = \frac{144}{169}

Use $\sin^2 \theta + \cos^2 \theta = 1$ .

\sin \theta = \frac{12}{13}

Take the positive root for Quadrant I.

Answer: $\frac{12}{13}$

\cos^2 \theta = 1 - \frac{1}{4} = \frac{3}{4}

Substitute.

\cos \theta = \pm\frac{\sqrt{3}}{2}

Take the root.

Cosine is negative in Quadrant III
Answer: $-\frac{\sqrt{3}}{2}$

\sec^2 \theta = 1 + 4 = 5

Use $1 + \tan^2 \theta = \sec^2 \theta$ .

\sec \theta = \sqrt{5}

Take the positive root.

Answer: $\sqrt{5}$

\csc^2 \theta = 1 + \frac{9}{16} = \frac{25}{16}

Use $1 + \cot^2 \theta = \csc^2 \theta$ .

\csc \theta = \frac{5}{4}

Take the positive root.

Answer: $\frac{5}{4}$

\cos \theta \cdot \frac{\sin \theta}{\cos \theta}

Rewrite tangent.

\sin \theta

Cancel $\cos \theta$ .

Answer: $\sin \theta$

1 + \tan^2 \theta

Replace $\sin^2 \theta + \cos^2 \theta$ with 1.

\sec^2 \theta

Use the second Pythagorean identity.

Answer: $\sec^2 \theta$

\frac{\tan^2 \theta}{\tan \theta}

Use $\sec^2 \theta - 1 = \tan^2 \theta$ .

\tan \theta

Cancel one factor.

Answer: $\tan \theta$

\frac{\cos \theta}{\sin \theta} \cdot \frac{1}{\cos \theta}

Rewrite.

\frac{1}{\sin \theta}

Cancel $\cos \theta$ .

\csc \theta

Use the reciprocal identity.

Answer: $\csc \theta$

\cos^2 \theta \cdot \frac{1}{\cos \theta}

Replace $1 - \sin^2 \theta$ with $\cos^2 \theta$ .

\cos \theta

Cancel one factor.

Answer: $\cos \theta$

10.

\tan^2 \theta = 1.25^2 - 1 = 1.5625 - 1 = 0.5625

Use $\tan^2 \theta = \sec^2 \theta - 1$ .

\tan \theta = 0.75

Take the positive root.

0.75 \cdot 12 = 9

Multiply by the run.

Answer: $\tan \theta = 0.75$ , so the roof rises 9 feet over a 12-foot run

FAQ

What are the fundamental trig identities?

They are the reciprocal, quotient and Pythagorean identities, often listed with the even-odd and cofunction identities. Together they connect all six trig functions.

What are the three Pythagorean identities?

$\sin^2 \theta + \cos^2 \theta = 1$ , $1 + \tan^2 \theta = \sec^2 \theta$ and $1 + \cot^2 \theta = \csc^2 \theta$ .

Why is sin²θ + cos²θ = 1?

On the unit circle, the point for $\theta$ is $(\cos \theta, \sin \theta)$ , and every point on the circle is 1 unit from the origin. By the Pythagorean theorem, $\cos^2 \theta + \sin^2 \theta = 1^2$ .

What is the difference between an identity and an equation?

An identity is true for every allowed value of the variable. A trig equation, like $\sin \theta = \frac{1}{2}$ , is true only for certain angles, which you solve for.

Summary

Reciprocal identities: $\csc \theta = \frac{1}{\sin \theta}$ , $\sec \theta = \frac{1}{\cos \theta}$ , $\cot \theta = \frac{1}{\tan \theta}$ .
Quotient identities: $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and $\cot \theta = \frac{\cos \theta}{\sin \theta}$ .
Pythagorean identities: $\sin^2 \theta + \cos^2 \theta = 1$ , $1 + \tan^2 \theta = \sec^2 \theta$ , $1 + \cot^2 \theta = \csc^2 \theta$ .
To simplify, rewrite in sine and cosine, use a Pythagorean identity, then cancel factors.
Use the quadrant to choose the sign after a square root.

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Next lesson: Verifying Trig Identities

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