SOLUTION: I'm working on trig identities. some are simple, some not. I see comments about working on one side only. why is that? Does it make a difference? if so, can you show a proble

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Question 1131350: I'm working on trig identities. some are simple, some not.
I see comments about working on one side only. why is that?
Does it make a difference?
if so, can you show a problem wehre is does make a difference?
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20059)   (Show Source): You can put this solution on YOUR website!
The reason is that you cannot assume from the start that what is 
given is really an identity, since that is what you are to prove.



Here is an example.  We start with a NON-identity:

sin(x) + cos(x) = -sin(x) - cos(x)    

Square both sides:

sinē(x) + 2sin(x)cos(x) + cosē(x) = sinē(x) + 2sin(x)cos(x) + cosē(x)

Both sides are exactly equal.

So if you were allowed to square both sides, you would be able to
prove that some NON-identities were identities!!!

Edwin
Answer by ikleyn(52798)   (Show Source): You can put this solution on YOUR website!
.

            It is interesting question. 


Let assume that you are given a task to prove that two expressions are identical:


    A = B.


Expressions can be trigonometric or algebraic - it does not matter.



You can make transformations over the left side and over the right side.



If your transformations of each side are equivalent, and if they result to an identity

    C = D,

then you may conclude that the original expressions A and B are identical.


    Simply because you can reverse your chain of logical conclusions (= transformations) from
     the identity  C = D  back to  A = B.



But if your transformations are  NOT equivalent and if they result to an identity

    C = D,

then you CAN NOT conclude that the original expressions are identical.


So, working with equivalent transformations, you can transform either side or even both sides - there is NO obstacles
for it and for validity of your final conclusion.

But if you work and use non-equivalent transformations on the way - then be careful - your final conclusion might be wrong.

-----------------

Regarding the example given by Edwin, he started from the hypothetical identity, squared both sides, obtained the identity -
but in this case he can not conclude that the original hypothetical identity is a real identity.

Because squaring  IS  NOT  an equivalent transformation: from    =  you can not conclude that  a = b.



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