math.gif (3247 bytes) VERIFYING TRIGONOMETRIC IDENTITIES

Copyright by Ingrid Stewart, Ph.D.  Please Send Questions and Comments to ingrid.stewart@csn.edu.

Fundamental Trigonometric Identities

Reciprocal Identities

                     

                    

Quotient Identities

          

Even-Odd Properties

         

           

          

Fundamental Pythagorean Identities

They are derived from the following unit circle

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Using the definitions of sine and cosine above, we find that in the unit circle and Therefore, we can rewrite the equation of the unit circle and , which is then called a Pythagorean Identity.

Other Fundamental Pythagorean Identities:      

If we divide every term of by cos x, we get another Pythagorean Identity

         

If we divide every term of by sin x, we get yet another Pythagorean Identity

         

Secondary Fundamental Pythagorean Identities (derived from manipulating the primary identities):

If we move the terms of as follows, we get yet another Pythagorean Identity

         

If we move the terms of as follows, we get yet another Pythagorean Identity

         

If we move the terms of as follows, we get yet another Pythagorean Identity

         

NOTE: YOU MUST MEMORIZE THE FUNDAMENTAL IDENTITIES!

Five (5) Ways to Manipulate Trigonometric Expressions and Many Examples!

Please note the following:

                

1.   Adding or subtracting trigonometric expressions:

(a)    

        

(b)    

         

 (c)    

         

(d)    

the common denominator is

to add the two fractions the number 1 has to be multiplied by cos x and the expression tan x by sin x, and finally we get

 (e)    

 both fractions already have the same denominator, therefore

(f)    

the common denominator is

to add the two fractions, the number 3 has to be multiplied by (tan x - sec x) and the number 5 by (tan x + sec x)

and multiplying out the numerator and combining like terms, we finally get

2.   Multiplying trigonometric expressions:

(a)    Use FOIL to expand the following expression

        

(b)     which is equal to

(c)      which is equal to

3.   Factoring trigonometric expressions:

(a)     Factor out the common factor:

 (b)     Use the Difference of Squares formula:

(c)     Factor the following expression just like the trinomial

4.   Separating rational trigonometric expressions:

(a)    

Note:  You cannot cancel out cos x in the fraction above.   Only factors can be canceled in rational expressions.

(b)    

(c)    

5.   Using fundamental identities to rewrite an expression:

(a)      rewrite

Given the fundamental identity ,

we can rewrite the expression as which is equal to

(b)     rewrite

Given the fundamental identity

we can rewrite the expression as

which is equal to

Here we can even do more!  How about we find the common denominator and then write the expression as a single fraction?

Given the fundamental identity , we can change the numerator to

 

to get which equals .

(c)      rewrite

Given the fundamental identity , we can change the numerator to

which is

Then we can cancel out the expression since it occurs both in the numerator and in the denominator to find which equals .

Now that we have learned how to manipulate trigonometric expressions,  we will use these concepts to verify trigonometric identities that are NOT considered to be "fundamental." 

Unfortunately, there are no well-defined rules to follow in verifying trigonometric identities, and the process is best learned by practice.  There is no right way or wrong way as long as you follow established algebra rules.  Most beginners will have success using a trial and error approach.

Besides algebraic rules, there is one rule that you must also follow, and that is that you must either change the right side to look like the left side or vice versa.  DO NOT WORK ON BOTH SIDES OF THE IDENTITY AT THE SAME TIME!

Guidelines for Verifying Trigonometric Identities

  • Try to change the side that appears to be the "more complicated" one first.  Note, that "more complicated" is in the eye of the beholder.
  • Use the five (5) manipulations discussed above separately or in combination with each other.
  • Let the appearance of the "other side" guide you in your decision-making process!

 

Example 01
Verify a Trigonometric Identity
Example 02
Verify a Trigonometric Identity
Example 03
Verify a Trigonometric Identity
Example 04
A trigonometric expression can be reduced to a constant. Find this constant.
Example 05
A trigonometric expression can be reduced to a single trigonometric ratio. Find this ratio.
Example 06
A trigonometric expression can be reduced to a difference of two trigonometric ratios. Find this difference.