Fundamental Trigonometric Identities

If you have spent any time working with trigonometric functions, you have probably noticed something: certain equations seem to be true no matter what angle you use. Plug in 30 degrees, it works. Plug in 45 degrees, it works. Plug in some random angle like 73.2 degrees, and somehow it still works. These are not coincidences. They are called identities, and they are among the most powerful tools in all of trigonometry.

So, if trig identities seem scary to you, then know that you are not alone. Many students feel overwhelmed when they first see a list of formulas to memorize. But here is the good news: you need to realize something. You already know how to use these identities. You may not believe it, but, really, you do. Every identity is just a relationship between trig functions that you have already been working with. Once you see where they come from, they stop being mysterious formulas and start being obvious facts.

Core Concepts

What Is an Identity?

An identity is an equation that is true for all valid input values. This is different from a regular equation, which is only true for specific values.

Compare these two statements:

• Regular equation: $\sin \theta = 0.5$ is only true when $\theta = 30°, 150°, 390°$, etc.
• Identity: $\sin^2 \theta + \cos^2 \theta = 1$ is true for every angle $\theta$

When we say an identity is true for “all valid values,” we mean all values where the expressions are defined. For example, $\tan \theta = \frac{\sin \theta}{\cos \theta}$ is an identity, but it is only valid when $\cos \theta \neq 0$ (so not at $90°, 270°$, etc., where tangent is undefined anyway).

The identities you will learn in this lesson fall into several families: reciprocal identities, quotient identities, Pythagorean identities, cofunction identities, and even-odd identities. Each family captures a different type of relationship between the trigonometric functions.

Reciprocal Identities

You already know that every trigonometric function has a reciprocal partner:

$$\csc \theta = \frac{1}{\sin \theta}$$

$$\sec \theta = \frac{1}{\cos \theta}$$

$$\cot \theta = \frac{1}{\tan \theta}$$

These are the reciprocal identities. They simply express the fact that cosecant, secant, and cotangent are defined as the reciprocals of sine, cosine, and tangent, respectively.

You can also flip these around:

$$\sin \theta = \frac{1}{\csc \theta}$$

$$\cos \theta = \frac{1}{\sec \theta}$$

$$\tan \theta = \frac{1}{\cot \theta}$$

Why are these useful? Because they let you convert between functions. If you have an expression involving $\csc \theta$ and you would rather work with $\sin \theta$ (which is often easier), you can substitute. This flexibility is the heart of simplifying trig expressions.

Quotient Identities

The quotient identities express tangent and cotangent in terms of sine and cosine:

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

$$\cot \theta = \frac{\cos \theta}{\sin \theta}$$

These come directly from the definitions. Remember that in a right triangle with angle $\theta$:

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
• $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

If you divide the first equation by the second:

$$\frac{\sin \theta}{\cos \theta} = \frac{\text{opposite}/\text{hypotenuse}}{\text{adjacent}/\text{hypotenuse}} = \frac{\text{opposite}}{\text{adjacent}} = \tan \theta$$

The quotient identities are perhaps the most frequently used of all the identities. Whenever you see tangent or cotangent and want to convert everything to sines and cosines (a common strategy), these are the formulas you will reach for.

Pythagorean Identities

These are the crown jewels of trigonometric identities. The first and most important is:

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

This identity comes directly from the Pythagorean theorem. If you place an angle $\theta$ in standard position on the unit circle, the point on the circle has coordinates $(\cos \theta, \sin \theta)$. Since this point lies on the unit circle (which has equation $x^2 + y^2 = 1$), we get $\cos^2 \theta + \sin^2 \theta = 1$.

From this fundamental identity, we can derive two more by dividing everything by either $\cos^2 \theta$ or $\sin^2 \theta$:

Dividing by $\cos^2 \theta$:

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

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

Or, rearranging: $1 + \tan^2 \theta = \sec^2 \theta$

Dividing by $\sin^2 \theta$:

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

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

So we have three Pythagorean identities:

1. $\sin^2 \theta + \cos^2 \theta = 1$
2. $1 + \tan^2 \theta = \sec^2 \theta$
3. $1 + \cot^2 \theta = \csc^2 \theta$

Each of these can be rearranged. For example, from the first identity:

• $\sin^2 \theta = 1 - \cos^2 \theta$
• $\cos^2 \theta = 1 - \sin^2 \theta$

These rearranged forms are equally important and come up constantly when simplifying expressions.

Cofunction Identities

The cofunction identities express the relationship between complementary angles (angles that add up to 90°). The prefix “co” in cosine, cotangent, and cosecant actually stands for “complementary.”

$$\sin(90° - \theta) = \cos \theta$$ $$\cos(90° - \theta) = \sin \theta$$

$$\tan(90° - \theta) = \cot \theta$$ $$\cot(90° - \theta) = \tan \theta$$

$$\sec(90° - \theta) = \csc \theta$$ $$\csc(90° - \theta) = \sec \theta$$

In radians, replace $90°$ with $\frac{\pi}{2}$:

$$\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta$$

Why do these work? Think about a right triangle. If one acute angle is $\theta$, the other acute angle is $90° - \theta$ (since the angles of a triangle sum to $180°$, and one angle is already $90°$). The side that is “opposite” to angle $\theta$ is “adjacent” to angle $90° - \theta$, and vice versa. This swapping of opposite and adjacent is exactly why sine and cosine are cofunctions of each other.

Even-Odd Identities

The even-odd identities tell you what happens when you plug in a negative angle:

Even function (symmetric about the y-axis): $$\cos(-\theta) = \cos \theta$$ $$\sec(-\theta) = \sec \theta$$

Odd functions (symmetric about the origin): $$\sin(-\theta) = -\sin \theta$$ $$\csc(-\theta) = -\csc \theta$$ $$\tan(-\theta) = -\tan \theta$$ $$\cot(-\theta) = -\cot \theta$$

Here is an easy way to remember which is which: cosine is even (the word “cosine” has an “e” in it, like “even”). Sine is odd (the word “sine” does not have an “e”). The other functions follow their reciprocals.

To visualize why these work, think about the unit circle. If you start at angle $\theta$ and reflect across the x-axis to get angle $-\theta$, the x-coordinate stays the same (so cosine is unchanged), but the y-coordinate becomes its negative (so sine changes sign).

Using Identities to Simplify Expressions

Now that you know the identities, the real skill is using them to simplify complicated expressions. The basic strategy is:

1. Convert everything to sines and cosines (using reciprocal and quotient identities)
2. Look for Pythagorean identity opportunities (especially $\sin^2 \theta + \cos^2 \theta = 1$)
3. Simplify fractions by combining terms and canceling
4. Convert back to other functions if needed

There is no single “right” method for every problem. Sometimes you convert to sines and cosines; sometimes you do not. With practice, you will develop intuition for which approach works best.

Verifying Identities

When you are asked to verify an identity, you need to show that the left side equals the right side. There are some important rules:

1. Work with only one side at a time. Pick the more complicated side and transform it into the simpler side. Do not work with both sides simultaneously (that would assume the identity is already true).

2. You cannot add, subtract, multiply, or divide both sides by something. Those operations are for solving equations, not verifying identities. You can only use substitutions and algebraic simplifications on one side.

3. Convert to sines and cosines is often a good starting strategy, especially when you are stuck.

4. Factor, combine fractions, or multiply by clever forms of 1 (like $\frac{\cos \theta}{\cos \theta}$) when needed.

Verifying identities is like a puzzle. You know both the starting point (one side) and the ending point (the other side). Your job is to find a path of valid substitutions that transforms one into the other.

Notation and Terminology

Term	Meaning	Example
Identity	Equation true for all valid values	$\sin^2 \theta + \cos^2 \theta = 1$
Verify	Show both sides are equal by transforming one into the other
Simplify	Reduce to a simpler form	$\frac{\sin \theta}{\cos \theta} = \tan \theta$
Cofunctions	Function pairs that differ by $90°$	sin/cos, tan/cot, sec/csc
Even function	$f(-x) = f(x)$	$\cos(-\theta) = \cos \theta$
Odd function	$f(-x) = -f(x)$	$\sin(-\theta) = -\sin \theta$

Examples

Key Properties and Rules

Summary of All Fundamental Identities

Reciprocal Identities:

Identity	Alternate Form
$\csc \theta = \frac{1}{\sin \theta}$	$\sin \theta = \frac{1}{\csc \theta}$
$\sec \theta = \frac{1}{\cos \theta}$	$\cos \theta = \frac{1}{\sec \theta}$
$\cot \theta = \frac{1}{\tan \theta}$	$\tan \theta = \frac{1}{\cot \theta}$

Quotient Identities:

Identity
$\tan \theta = \frac{\sin \theta}{\cos \theta}$
$\cot \theta = \frac{\cos \theta}{\sin \theta}$

Pythagorean Identities:

Primary Form	Alternate Forms
$\sin^2 \theta + \cos^2 \theta = 1$	$\sin^2 \theta = 1 - \cos^2 \theta$, $\cos^2 \theta = 1 - \sin^2 \theta$
$1 + \tan^2 \theta = \sec^2 \theta$	$\tan^2 \theta = \sec^2 \theta - 1$, $\sec^2 \theta - \tan^2 \theta = 1$
$1 + \cot^2 \theta = \csc^2 \theta$	$\cot^2 \theta = \csc^2 \theta - 1$, $\csc^2 \theta - \cot^2 \theta = 1$

Cofunction Identities (with $90° = \frac{\pi}{2}$):

Identity	Identity
$\sin(90° - \theta) = \cos \theta$	$\cos(90° - \theta) = \sin \theta$
$\tan(90° - \theta) = \cot \theta$	$\cot(90° - \theta) = \tan \theta$
$\sec(90° - \theta) = \csc \theta$	$\csc(90° - \theta) = \sec \theta$

Even-Odd Identities:

Even (unchanged)	Odd (sign changes)
$\cos(-\theta) = \cos \theta$	$\sin(-\theta) = -\sin \theta$
$\sec(-\theta) = \sec \theta$	$\tan(-\theta) = -\tan \theta$
	$\csc(-\theta) = -\csc \theta$
	$\cot(-\theta) = -\cot \theta$

Strategies for Simplifying and Verifying

1. Convert to sines and cosines when you are unsure what to do
2. Look for Pythagorean identity patterns, especially $1 - \sin^2 \theta$, $1 - \cos^2 \theta$, $\sec^2 \theta - 1$, etc.
3. Factor when you see difference of squares like $1 - \cos^2 \theta = (1-\cos\theta)(1+\cos\theta)$
4. Combine fractions over a common denominator
5. Multiply by clever forms of 1 like $\frac{1 + \cos \theta}{1 + \cos \theta}$ to create useful forms
6. Work with the more complicated side when verifying

Real-World Applications

Simplifying Physics Equations

Physics is full of trigonometric expressions that need simplifying. For example, when analyzing projectile motion, you might encounter expressions involving both sine and cosine of the launch angle. Using identities like $\sin^2 \theta + \cos^2 \theta = 1$ can dramatically simplify these calculations, reducing complex expressions to elegant results.

Signal Processing

In electrical engineering and signal processing, signals are often represented as combinations of sine and cosine waves. The identities allow engineers to convert between different representations, combine signals, and analyze their properties. For instance, converting a signal written in terms of $\sin \theta$ and $\cos \theta$ into one using only cosine (or only sine) uses these fundamental identities.

Engineering Calculations

Structural engineers analyzing forces on bridges, buildings, and other structures frequently decompose forces into components using trigonometry. The Pythagorean identities help verify that the component forces are consistent with the total force, serving as a check on calculations.

Computer Graphics Optimizations

When rendering 3D graphics, computers perform millions of trigonometric calculations per second. Game developers and graphics programmers use identities to replace expensive operations with simpler ones. For example, instead of computing both $\sin \theta$ and $\cos \theta$ separately, you can compute one and use the Pythagorean identity to find the other (which is faster). These optimizations, applied millions of times, make real-time 3D graphics possible.

Navigation and Surveying

GPS systems and surveying equipment use trigonometry extensively. When calculating distances and angles on the Earth’s surface, the identities help convert between different coordinate systems and simplify the complex equations involved in accounting for the Earth’s curvature.

Self-Test Problems

Summary

• An identity is an equation that is true for all valid input values, not just specific solutions
• Reciprocal identities express csc, sec, and cot as reciprocals: $\csc \theta = \frac{1}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, $\cot \theta = \frac{1}{\tan \theta}$
• Quotient identities express tangent and cotangent in terms of sine and cosine: $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and $\cot \theta = \frac{\cos \theta}{\sin \theta}$
• The Pythagorean identities are the most important:
  • $\sin^2 \theta + \cos^2 \theta = 1$ (fundamental)
  • $1 + \tan^2 \theta = \sec^2 \theta$ (divide by $\cos^2 \theta$)
  • $1 + \cot^2 \theta = \csc^2 \theta$ (divide by $\sin^2 \theta$)
• Cofunction identities relate functions at complementary angles: $\sin(90° - \theta) = \cos \theta$
• Even-odd identities describe behavior with negative angles: cosine and secant are even; the others are odd
• To simplify expressions, convert to sines and cosines, look for Pythagorean patterns, and factor when possible
• To verify identities, work with one side only and transform it into the other side
• These identities have practical applications in physics, engineering, signal processing, and computer graphics