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Verifying Trigonometric Identities

Date: 07/12/2002 at 08:10:05
From: Dianne Graham
Subject: How do you verify a Trigonometry Identity?

How do you verify a Trigonometry Identity?  I looked in my textbook 
and the directions were not clear.


Date: 07/12/2002 at 17:24:22
From: Doctor Ian
Subject: Re: How do you verify a Trigonometry Identity?

Hi Diane,

The idea is to manipulate one or both sides of the identity until 
you have something that is obviously true, e.g., sin(x) = sin(x). 

It might be instructive to see how you can _create_ an identity.  
Start with something like 

  sin(x) = sin(x)

Now, csc(x) = 1/sin(x), so we can write

  sin(x) = 1/csc(x)

And tan(x) = sin(x)/cos(x), so we can write

  tan(x)cos(x) = 1/csc(x)

And cos^2(x) + sin^2(x) = 1, so we can write

                 sin^2(x) + cos^2(x)
  tan(x)cos(x) = -------------------
                       csc(x)


                 sin^2(x)   cos^2(x)
               = -------- + --------
                  csc(x)     csc(x)

       
                            cos^2(x)
               = sin^3(x) + --------
                             csc(x)

Now, this isn't a very elegant identity, but it illustrates 
the point:  By starting with a trivially true statement, and 
exchanging simple expressions for more complicated equivalent 
expressions, you can build up a statement that must be true, but 
isn't obviously true. 

It's very much like what you do to set up an equation to be 
solved in algebra:

    http://mathforum.org/library/drmath/view/57265.html 

Verifying an identity is a matter of going in the opposite 
direction.  Usually you start by converting everything to sines 
and cosines:  

                            cos^2(x)
  tan(x)cos(x) = sin^3(x) + --------
                             csc(x)


  sin(x)                    cos^2(x)
  ------cos(x) = sin^3(x) + --------
  cos(x)                    1/sin(x)


And then you use the tricks - e.g., factoring, cancellation - 
that you were supposed to have learned in algebra:

        sin(x) = sin^3(x) + sin(x)cos^2(x)

               = sin(x)(sin^2(x) + cos^2(x))

        sin(x) = sin(x)


Often the key to verifying an identity is to recognize that 
something like 

  sin(x)cos(y) + cos(x)sin(y),

can be replaced by 

  sin(x+y)
  
There's no magic formula for doing these kinds of problems.  It 
normally involves a lot of trial and error, since you're 
essentially untying a knot that someone else tied, often just 
because he thought of a tricky way to tie one.  

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Trigonometry

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