Pythagorean Identities
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When studying the unit circle, it was observed that a point on the unit circle (the vertex of the right triangle) can be represented by the coordinates . 

Since the legs of the right triangle in the unit circle have the values of and , the Pythagorean Theorem can be used to obtain .

This well-known equation is called a Pythagorean Identity. 
The value of is immaterial. 

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

  Start with the first Pythagorean Identity.

 Divide each term by .

 Remember:

 Reduce and substitute.
 

The second Pythagorean Identity is:

 

Now, for the third equation:

  Start with the first Pythagorean Identity.

 Divide each term by .

 Remember:

 Reduce and substitute.


  The third Pythagorean Identity is:

 

Many times it will be necessary to use a "version" of these Pythagorean Identities.
Be on the look-out for these variations.
Pythagorean Identity Variations
                 
                           
                           

 

Examples:

The Pythagorean Identities may be used to find missing trigonometric values.

1. 


find the value of the other trig functions.

Of course, this problem can also be solved without the use of the Pythagorean Identities.
 

 



 

 

 

 

 

 

A more widely known use of the Pythagorean Identities is to help simplify trigonometric expressions.

2.         
  Start by factoring:

Utilize this substitution:

 

 

Pythagorean Identities are also helpful in simplifying trigonometric expressions to create a factorable expression.

3.      
  Start by substituting:

This identity will help:

 

Such processes, as seen here, will also prove valuable when solving trigonometric equations.

 

How to use your
TI-83+/84+ graphing calculator  to verify trigonometric identities.
Click calculator.