Math Core

An identity is an equation that is true for every value of the variable where both sides are defined. You have already met a few trig identities without calling them that, such as $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ . This lesson collects the basic ones in one place. They are the tools you will use for the rest of the unit to rewrite, simplify and evaluate trig expressions.

Identities versus equations

An equation like $\sin\theta = \dfrac{1}{2}$ is true only for certain angles, such as $30^\circ$ and $150^\circ$ . An identity like $\sin^2\theta + \cos^2\theta = 1$ is true for every angle. Solving an equation means finding the angles that work. Using an identity means swapping one expression for another that is always equal to it.

Definition

Trigonometric identity

A trigonometric identity is an equation involving trig functions that is true for every value of the variable for which both sides are defined.

The phrase "where both sides are defined" matters. For example, $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ is an identity even though neither side exists at $\theta = 90^\circ$ . We only compare the sides where they make sense.

Reciprocal and quotient identities

These come straight from the definitions of the six trig functions on the unit circle, where the point on the terminal side is $(\cos\theta, \sin\theta)$ .

\csc\theta = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta}

\tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \cot\theta = \frac{\cos\theta}{\sin\theta}

Because of these, every trig expression can be written using only sine and cosine. That is often the first move when an expression looks messy.

The Pythagorean identities

The point $(\cos\theta, \sin\theta)$ lies on the unit circle $x^2 + y^2 = 1$ . Substituting gives the most important identity in trigonometry:

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

The notation $\sin^2\theta$ means $(\sin\theta)^2$ .

Divide every term by $\cos^2\theta$ (where $\cos\theta \ne 0$ ):

\frac{\cos^2\theta}{\cos^2\theta} + \frac{\sin^2\theta}{\cos^2\theta} = \frac{1}{\cos^2\theta} \quad\Longrightarrow\quad 1 + \tan^2\theta = \sec^2\theta.

Divide instead by $\sin^2\theta$ (where $\sin\theta \ne 0$ ):

\frac{\cos^2\theta}{\sin^2\theta} + \frac{\sin^2\theta}{\sin^2\theta} = \frac{1}{\sin^2\theta} \quad\Longrightarrow\quad \cot^2\theta + 1 = \csc^2\theta.

The Pythagorean identities

\sin^2\theta + \cos^2\theta = 1 \qquad 1 + \tan^2\theta = \sec^2\theta \qquad 1 + \cot^2\theta = \csc^2\theta

Each one can be rearranged. For example, $\sin^2\theta = 1 - \cos^2\theta$ and $\sec^2\theta - 1 = \tan^2\theta$ .

The graph below shows $y = \sin^2 x$ and $y = \cos^2 x$ . Wherever one is high, the other is low, and at every $x$ their heights add to exactly $1$ .

The graphs of sin²x and cos²x. At every x, their heights add to 1 (the dashed line).Open in grapher →

Even-odd and cofunction identities

Reflecting an angle across the $x$ -axis turns $\theta$ into $-\theta$ . The $x$ -coordinate stays the same and the $y$ -coordinate changes sign, so

\cos(-\theta) = \cos\theta \qquad \sin(-\theta) = -\sin\theta \qquad \tan(-\theta) = -\tan\theta.

Cosine (and its reciprocal secant) is even. Sine, tangent, and their reciprocals are odd.

In a right triangle, the two acute angles add to $90^\circ$ , and the side opposite one angle is adjacent to the other. That gives the cofunction identities:

\sin(90^\circ - \theta) = \cos\theta \qquad \cos(90^\circ - \theta) = \sin\theta \qquad \tan(90^\circ - \theta) = \cot\theta

In radians, replace $90^\circ$ with $\dfrac{\pi}{2}$ . The "co" in cosine literally means "sine of the complement."

Finding values from one known value

If you know one trig value and the quadrant, the Pythagorean identity gives the rest.

Worked example: All six from one

Suppose $\sin\theta = -\dfrac{5}{13}$ and $\theta$ is in Quadrant III. Find the other five trig values.

Use $\cos^2\theta = 1 - \sin^2\theta$ :

\cos^2\theta = 1 - \frac{25}{169} = \frac{144}{169}, \qquad \cos\theta = \pm\frac{12}{13}.

Cosine is negative in Quadrant III, so $\cos\theta = -\dfrac{12}{13}$ . Now use the quotient and reciprocal identities:

\tan\theta = \frac{-5/13}{-12/13} = \frac{5}{12}, \qquad \csc\theta = -\frac{13}{5}, \qquad \sec\theta = -\frac{13}{12}, \qquad \cot\theta = \frac{12}{5}.

Common mistake

Taking a square root always gives a $\pm$ . The identity alone can't tell you the sign. Use the quadrant: all positive in QI, only sine and cosecant in QII, only tangent and cotangent in QIII, only cosine and secant in QIV.

Simplifying expressions

To simplify, look for a Pythagorean pattern, or rewrite everything in sines and cosines and then clean up the fractions.

Worked example: Simplify using a Pythagorean identity

Simplify $\dfrac{1 - \cos^2 x}{\sin x}$ .

Replace the numerator with $\sin^2 x$ :

\frac{1 - \cos^2 x}{\sin x} = \frac{\sin^2 x}{\sin x} = \sin x.

Worked example: Rewrite in sines and cosines

Simplify $\tan x \csc x \cos x$ .

\begin{aligned} \tan x \csc x \cos x &= \frac{\sin x}{\cos x} \cdot \frac{1}{\sin x} \cdot \cos x \\ &= \frac{\sin x \cos x}{\sin x \cos x} \\ &= 1 \end{aligned}

The whole product is the constant $1$ (wherever it is defined).

Worked example: Factoring first

Simplify $\sin^2 x \sec^2 x - \sec^2 x$ .

Both terms share a factor of $\sec^2 x$ , so factor it out first.

\begin{aligned} \sin^2 x \sec^2 x - \sec^2 x &= \sec^2 x\,(\sin^2 x - 1) \\ &= \sec^2 x\,(-\cos^2 x) \\ &= \frac{1}{\cos^2 x}\cdot(-\cos^2 x) \\ &= -1 \end{aligned}

Tip

Check a simplification by plugging in an angle you know, such as $x = 30^\circ$ or $x = 45^\circ$ . If the original and the simplified form give different numbers, something went wrong.

Practice

Practice 1

Which expression is equal to $\sec^2\theta - \tan^2\theta$ ?

Practice 2

If $\sin\theta = \dfrac{3}{5}$ and $\theta$ is in Quadrant II, what is $\cos\theta$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

If $\cos\theta = \dfrac{8}{17}$ and $\theta$ is in Quadrant IV, what is $\tan\theta$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Which expression is equal to $\cot x \sin x$ ?

Practice 5

Which expression is equal to $\sin(-x)\csc x$ ?

Practice 6

Which expression is equal to $\dfrac{\csc^2 x - 1}{\csc^2 x}$ ?

Practice 7

The expression $(1 - \sin x)(1 + \sin x)\sec^2 x$ equals a constant wherever it is defined. What is that constant?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

If $\tan\theta = 3$ and $\theta$ is in Quadrant III, what is $\sec\theta$ ? Give an exact answer.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.