Using this first Pythagorean Identity, two additional Pythagorean Identities can be created.
The second Pythagorean Identity is:

Start with the first Pythagorean Identity.
Divide each term by .

Remember:
Reduce and Substitute.

The third Pythagorean Identity is:

Start with the first Pythagorean Identity.

Divide each term by .

Remember:
Reduce and Substitute.

How to Cancel Identities

If you have the opposite trigonometric functions, they cancel each other out to equal one.
tanx cotx= 1 ;because cotx=1/tanx then tanx times 1/tanx= tanx/tanx=1
cosx secx= 1;because secx=1/cosx then cosx times 1/cosx= cosx/cosx=1 sinx cscx= 1;because cscx=1/sinx then sinx times 1/sinx= sinx/sinx=1

These three formulas can be algebraically altered in various ways to make other equations, as listed below:

sin²x + cos²x = 1 ---► 1 - sin²x = cos²x

sin²x + cos²x = 1 ---► 1 - cos²x = sin²x

tan²x + 1 = sec²x ---► sec²x - tan²x = 1

tan²x + 1 = sec²x ---► sec²x - 1 = tan²x

1 + cot²x = csc²x ---► csc²x - cot²x = 1

1 + cot²x = csc²x ---► csc²x - 1 = cot²x

Quotient Identities

tan θ =

sin θ
cos θ

cot θ =

cos θ
sin θ

This identity is used for changing tangent/cotangent into terms of sine and cosine, or vice-versa.

Reciprocal Identities

sin θ

=

1csc θ

csc θ

=

1sin θ

cos θ

=

1sec θ

sec θ

=

1cos θ

tan θ

=

1cot θ

cot θ

=

1tan θ

When multiplying a trigonometric function by its inverse, the result is 1. For instance...

sin(θ) x csc(θ) = 1

As a numerical example, 1/2 x 2/1 = 1. In this case, the same principle applies.

This can also be used to move a denominator to the numerator (or vise-versa).
Ex: tan(x)/sec(x) can be changed to cos(x) tan(x).

Odd-Even Identities

sin(-x)= -sinx

cos(-x) = cosx

tan(-x) = -tanx

csc(-x)= -cscx

sec(-x)= secx

cot(-x)= -cotx

Odd functions have opposite end behavior.
Sine, cosecant, tangent, and co tangent are odd functions.

Even functions have the same end behavior. Cosine and secant are even functions

## Table of Contents

Trigonometric Identities## Pythagorean Identities

This is a

Pythagorean Identity:Using this first Pythagorean Identity, two additional Pythagorean Identities can be created.

The second

Pythagorean Identityis:Divide each term by .

Remember:

Reduce and Substitute.

The third

Pythagorean Identityis:Divide each term by .

Remember:

Reduce and Substitute.

How to Cancel Identitiestanx cotx= 1 ;because cotx=1/tanx then tanx times 1/tanx= tanx/tanx=1

cosx secx= 1;because secx=1/cosx then cosx times 1/cosx= cosx/cosx=1

sinx cscx= 1;because cscx=1/sinx then sinx times 1/sinx= sinx/sinx=1

These three formulas can be algebraically altered in various ways to make other equations, as listed below:

sin²x + cos²x = 1 ---► 1 - sin²x = cos²xsin²x + cos²x = 1 ---► 1 - cos²x = sin²xtan²x + 1 = sec²x ---► sec²x - tan²x = 1tan²x + 1 = sec²x ---► sec²x - 1 = tan²x1 + cot²x = csc²x ---► csc²x - cot²x = 11 + cot²x = csc²x ---► csc²x - 1 = cot²xQuotient Identitiessin θcos θ

cos θsin θ

Reciprocal Identities=1csc θ=1sin θ=1sec θ=1cos θ=1cot θ=1tan θAs a numerical example, 1/2 x 2/1 = 1. In this case, the same principle applies.

This can also be used to move a denominator to the numerator (or vise-versa).

Ex: tan(x)/sec(x) can be changed to cos(x) tan(x).

## Odd-Even Identities

Sine, cosecant, tangent, and co tangent are odd functions.

Even functions have the same end behavior.

Cosine and secant are even functions

## Videos

Fundamental trigonometric identities

An Easy to Understand Lesson on Identities