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Verify the Identity

Date: 04/17/2003 at 20:30:56
From: April
Subject: How do you work this problem?

Verify the identity:      csc x
                      ------------- = cos x
                      tan x + cot x


Date: 04/18/2003 at 16:46:15
From: Doctor Link
Subject: Re: How do you work this problem?

Hi April, 

I'm glad you chose to write about verifying identities, because that 
happens to be something that I really enjoy.

The first thing that you should do when trying to verify an identity 
is have a list of all the most common identities beside you. You can 
refer to the following FAQ to view such a list: 

   Trigonometry Formulas - Dr. Math FAQ
   http://mathforum.org/dr.math/faq/formulas/faq.trig.html   

Equipped with this knowledge you can now begin to try to verify this 
identity. What I find to be a very useful method is to rewrite the 
more complicated side (the side with more terms in it - in this case 
the left side) in terms of sine and cosine. This can be especially 
effective when the simpler side is mostly in terms of sine and 
cosine, as in this case.

Another effective method is to look for parts of the problem where, 
using identities, you can cancel things out. For example, suppose you
are trying to verify the identity:

                       2       2
  (csc x)*(sin x) = sin x + cos x

Since we know that csc x is the reciprocal of sin x, we can rewrite 
the left side as:
  
    1   
  ----- * (sin x)
  sin x 

which is, of course, just 1. To finish this problem you can rewrite 
the right side as 1 because of the Pythagorean identity:

     2       2
  sin x + cos x = 1
  
Now let's go back to your problem. To rewrite the csc x part in terms 
of sine or cosine you simply make use of the fact that:
   
     1 
   -----
   sin x
 
is its reciprocal.
 
Now how could you change the tan x and the cot x to only sines and 
cosines? Try to use the general techniques I mentioned above.

If you are still having trouble feel free to write back and I will 
assist you further. Thanks again for writing in!
                             
- Doctor Link, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Trigonometry

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