Table of Contents
In trigonometry, the Pythagorean identities are special equations that connect the three basic trigonometric functions: sine, cosine, and tangent. They are called “Pythagorean” because they come directly from the Pythagorean theorem applied to a right triangle or to the unit circle.
This chapter focuses on:
- Stating the main Pythagorean identities.
- Showing where they come from.
- Learning how to rearrange and use them to simplify expressions and solve simple equations.
You should already be familiar with sine, cosine, tangent, and the unit circle idea from earlier chapters in this section.
The Fundamental Pythagorean Identity
The most important identity is:
$$
\sin^2 \theta + \cos^2 \theta = 1
$$
Here $\sin^2 \theta$ is a shorthand for $(\sin \theta)^2$, and similarly for $\cos^2 \theta$.
This identity holds for every real angle $\theta$ for which sine and cosine are defined.
Geometric idea (right triangle)
In a right triangle with angle $\theta$ and hypotenuse of length $1$:
- The side opposite $\theta$ has length $\sin \theta$.
- The side adjacent to $\theta$ has length $\cos \theta$.
- The hypotenuse has length $1$.
By the Pythagorean theorem:
$$
(\text{opposite})^2 + (\text{adjacent})^2 = (\text{hypotenuse})^2
$$
becomes
$$
(\sin \theta)^2 + (\cos \theta)^2 = 1^2 \quad\Rightarrow\quad \sin^2 \theta + \cos^2 \theta = 1.
$$
Unit circle view
On the unit circle, a point corresponding to angle $\theta$ has coordinates $(\cos \theta, \sin \theta)$. The distance from the origin to this point is $1$, so:
$$
(\cos \theta)^2 + (\sin \theta)^2 = 1^2.
$$
Again, this gives $\sin^2 \theta + \cos^2 \theta = 1$.
Rearranging the Fundamental Identity
You can solve the fundamental identity for $\sin^2 \theta$ or $\cos^2 \theta$:
From
$$
\sin^2 \theta + \cos^2 \theta = 1
$$
subtract $\sin^2 \theta$ from both sides:
$$
\cos^2 \theta = 1 - \sin^2 \theta.
$$
Or subtract $\cos^2 \theta$ from both sides:
$$
\sin^2 \theta = 1 - \cos^2 \theta.
$$
These forms are especially useful when you want to:
- Replace $\sin^2 \theta$ in an expression with $1 - \cos^2 \theta$, or
- Replace $\cos^2 \theta$ with $1 - \sin^2 \theta$,
in order to simplify an expression so that it uses only sine or only cosine.
Example of use
Simplify $\sin^2 \theta + \sin^2 \theta \cos^2 \theta$.
Factor out $\sin^2 \theta$:
$$
\sin^2 \theta + \sin^2 \theta \cos^2 \theta = \sin^2 \theta(1 + \cos^2 \theta).
$$
Now use $1 = \sin^2 \theta + \cos^2 \theta$:
$$
1 + \cos^2 \theta = (\sin^2 \theta + \cos^2 \theta) + \cos^2 \theta = \sin^2 \theta + 2\cos^2 \theta.
$$
So
$$
\sin^2 \theta(1 + \cos^2 \theta) = \sin^2 \theta(\sin^2 \theta + 2\cos^2 \theta).
$$
This is not “simpler” in every sense, but it shows how the identity lets you rewrite expressions in different equivalent forms.
A more typical simplification: rewrite $\sin^2 \theta$ in terms of $\cos^2 \theta$:
$$
3 - 2\sin^2 \theta = 3 - 2(1 - \cos^2 \theta) = 3 - 2 + 2\cos^2 \theta = 1 + 2\cos^2 \theta.
$$
Pythagorean Identities with Tangent and Secant
Starting from
$$
\sin^2 \theta + \cos^2 \theta = 1,
$$
you can derive identities involving tangent and secant by dividing each term by $\cos^2 \theta$ (where $\cos \theta \neq 0$):
$$
\frac{\sin^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta}.
$$
Now simplify each fraction:
- $\dfrac{\sin^2 \theta}{\cos^2 \theta} = \left(\dfrac{\sin \theta}{\cos \theta}\right)^2 = \tan^2 \theta$,
- $\dfrac{\cos^2 \theta}{\cos^2 \theta} = 1$,
- $\dfrac{1}{\cos^2 \theta} = \sec^2 \theta$.
So we get the second Pythagorean identity:
$$
\tan^2 \theta + 1 = \sec^2 \theta.
$$
You can rearrange it as:
$$
\tan^2 \theta = \sec^2 \theta - 1
\quad\text{or}\quad
1 = \sec^2 \theta - \tan^2 \theta.
$$
These are useful for rewriting powers of tangent in terms of secant and vice versa.
Example of use
Simplify $\sec^2 \theta - \tan^2 \theta$.
Using the identity $\tan^2 \theta + 1 = \sec^2 \theta$:
$$
\sec^2 \theta - \tan^2 \theta = (\tan^2 \theta + 1) - \tan^2 \theta = 1.
$$
Pythagorean Identities with Cotangent and Cosecant
Similarly, divide the fundamental identity by $\sin^2 \theta$ (where $\sin \theta \neq 0$):
$$
\frac{\sin^2 \theta}{\sin^2 \theta} + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{1}{\sin^2 \theta}.
$$
Simplify each fraction:
- $\dfrac{\sin^2 \theta}{\sin^2 \theta} = 1$,
- $\dfrac{\cos^2 \theta}{\sin^2 \theta} = \left(\dfrac{\cos \theta}{\sin \theta}\right)^2 = \cot^2 \theta$,
- $\dfrac{1}{\sin^2 \theta} = \csc^2 \theta$.
This gives the third Pythagorean identity:
$$
1 + \cot^2 \theta = \csc^2 \theta.
$$
Again, you can rearrange it:
$$
\cot^2 \theta = \csc^2 \theta - 1
\quad\text{or}\quad
1 = \csc^2 \theta - \cot^2 \theta.
$$
Example of use
Simplify $\csc^2 \theta - 1$.
From $1 + \cot^2 \theta = \csc^2 \theta$:
$$
\csc^2 \theta - 1 = \cot^2 \theta.
$$
Summary of the Pythagorean Identities
It is helpful to keep all three in one place:
- Fundamental identity:
$$
\sin^2 \theta + \cos^2 \theta = 1.
$$ - Tangent–secant identity (valid where $\cos \theta \neq 0$):
$$
\tan^2 \theta + 1 = \sec^2 \theta.
$$ - Cotangent–cosecant identity (valid where $\sin \theta \neq 0$):
$$
1 + \cot^2 \theta = \csc^2 \theta.
$$
Each of the last two comes from the first by dividing through by $\cos^2 \theta$ or $\sin^2 \theta$.
Using Pythagorean Identities to Find Missing Values
The identities allow you to find one trigonometric function when you know another, up to sign (positive or negative). The sign depends on the quadrant of the angle.
Example 1: Given sine, find cosine
Suppose $\sin \theta = \dfrac{3}{5}$ and $\theta$ is in the first quadrant.
Use $\sin^2 \theta + \cos^2 \theta = 1$:
$$
\left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1
\quad\Rightarrow\quad
\frac{9}{25} + \cos^2 \theta = 1.
$$
Subtract $\frac{9}{25}$:
$$
\cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25}.
$$
So
$$
\cos \theta = \pm \frac{4}{5}.
$$
Because $\theta$ is in the first quadrant, cosine is positive, so $\cos \theta = \dfrac{4}{5}$.
Example 2: Given secant, find tangent
Suppose $\sec \theta = 5$ and $\theta$ is in the fourth quadrant.
Use $\tan^2 \theta + 1 = \sec^2 \theta$:
$$
\tan^2 \theta + 1 = 5^2 = 25
\quad\Rightarrow\quad
\tan^2 \theta = 24.
$$
So
$$
\tan \theta = \pm \sqrt{24} = \pm 2\sqrt{6}.
$$
In the fourth quadrant, tangent is negative, so $\tan \theta = -2\sqrt{6}$.
Using Pythagorean Identities to Simplify Expressions
The Pythagorean identities are often used to turn expressions into simpler or more convenient forms.
Here are a few basic patterns:
- Replace $\sin^2 \theta$ by $1 - \cos^2 \theta$ to express everything in terms of cosine.
- Replace $\cos^2 \theta$ by $1 - \sin^2 \theta$ to express everything in terms of sine.
- Replace $\sec^2 \theta$ by $1 + \tan^2 \theta$, or vice versa.
- Replace $\csc^2 \theta$ by $1 + \cot^2 \theta$, or vice versa.
Example 3: Simplify an expression with $\sec^2$
Simplify $\sec^2 \theta - 3$ in terms of $\tan \theta$.
Using $\sec^2 \theta = 1 + \tan^2 \theta$:
$$
\sec^2 \theta - 3 = (1 + \tan^2 \theta) - 3 = \tan^2 \theta - 2.
$$
Example 4: Check an identity
Verify that
$$
\frac{\cos^2 \theta}{1 - \sin^2 \theta} = 1
$$
for angles where both sides are defined.
Using $1 - \sin^2 \theta = \cos^2 \theta$:
$$
\frac{\cos^2 \theta}{1 - \sin^2 \theta}
= \frac{\cos^2 \theta}{\cos^2 \theta} = 1.
$$
So the identity holds wherever $\cos \theta \neq 0$ (to avoid division by zero).
Common Pitfalls
When working with Pythagorean identities, watch out for these issues:
- Forgetting that $\sin^2 \theta$ means $(\sin \theta)^2$, not $\sin(\theta^2)$.
- Ignoring domain restrictions: when you divide by $\sin^2 \theta$ or $\cos^2 \theta$, you must assume they are not zero.
- Losing track of signs: from $\cos^2 \theta = \dfrac{16}{25}$ you get $\cos \theta = \pm \dfrac{4}{5}$. The correct sign depends on the quadrant of $\theta$.
- Trying to take a “square root of both sides” of an identity without considering that this can discard negative values or introduce extraneous ones; it is usually safer to work directly with the squared identity.
The Pythagorean identities are among the most frequently used identities in trigonometry. They will appear again in simplifying expressions, solving trigonometric equations, proving further trigonometric identities, and in calculus when integrating or differentiating trigonometric functions.