Trig Identities: sin²θ + cos²θ = 1 and the Double-Angle Formulas

Q: Why is sin²θ + cos²θ = 1?

On the unit circle, the point (cos θ, sin θ) is distance 1 from the origin. Pythagoras gives cos^2θ + sin^2θ = 1^2.

Where the Pythagorean identity comes from, how to use it to find one trig value from another, the quotient identity, the double-angle formulas, and how to prove an identity.

3 min read Supports 3 lessons

An equation like $2x + 1 = 7$ is true for one value of $x$ . An identity is true for every value. Trig identities let you swap one expression for another that is easier to work with.

The Pythagorean identity

On the unit circle, the point $(\cos\theta, \sin\theta)$ is always distance $1$ from the origin. Its legs are $\cos\theta$ and $\sin\theta$ and its hypotenuse is $1$ , so Pythagoras gives the most important identity in trigonometry.

Pythagorean identity

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

Watch out

$\sin^2\theta$ means $(\sin\theta)^2$ , not $\sin(\theta^2)$ .

Using it

$\sin\theta = \tfrac35, \;\; \theta \text{ acute}$
Find $\cos\theta$ .

1/4

Tip

This is the $3$ – $4$ – $5$ triangle in disguise. Sketching it is often the quickest way.

The quotient identity

Quotient identity

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

Since $\tan = \frac{\text{opp}}{\text{adj}}$ , divide the top and bottom by the hypotenuse.

Dividing the Pythagorean identity by $\cos^2\theta$ gives a second version: $\tan^2\theta + 1 = \sec^2\theta$ , where $\sec\theta = \frac{1}{\cos\theta}$ .

Double-angle formulas

Double-angle identities

\sin 2\theta = 2\sin\theta\cos\theta \qquad \cos 2\theta = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta

Worked example

If $\sin\theta = \frac35$ and $\theta$ is acute, find $\sin 2\theta$ .

1
From above, $\cos\theta = \frac45$ .
$\sin 2\theta = 2 \cdot \tfrac35 \cdot \tfrac45$

Watch out

$\sin 2\theta \neq 2\sin\theta$ . Check with $\theta = 90°$ : $\sin 180° = 0$ , but $2\sin 90° = 2$ .

Proving identities

To prove an identity, start from one side and transform it step by step into the other. Do not treat it as an equation and do the same thing to both sides: that assumes what you are trying to prove.

Worked example

Prove that $(1 - \cos\theta)(1 + \cos\theta) = \sin^2\theta$ .

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Expand the left side: a difference of two squares.
$1 - \cos^2\theta$

Key idea

Calculus leans on these identities, for example to rewrite $\sin^2 x$ as $\frac{1 - \cos 2x}{2}$ so that it can be integrated.

Common questions

Why is sin²θ + cos²θ = 1?: On the unit circle, the point $(\cos\theta, \sin\theta)$ is distance $1$ from the origin. Pythagoras gives $\cos^2\theta + \sin^2\theta = 1^2$ .
What is the difference between an equation and an identity?: An equation is true for particular values, which you solve for. An identity is true for every value of the variable.

Practise trig identities

The Trig Identities lesson has worked examples and unlimited practice that adapts to you.

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Common questions

Why is sin²θ + cos²θ = 1?

On the unit circle, the point $(\cos\theta, \sin\theta)$ is distance $1$ from the origin. Pythagoras gives $\cos^2\theta + \sin^2\theta = 1^2$ .

What is the difference between an equation and an identity?

An equation is true for particular values, which you solve for. An identity is true for every value of the variable.