Question 1131350
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            It is interesting question.



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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 <U>equivalent</U>, and if they result to an identity

    C = D,

then you <U>may conclude</U> 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  <U>NOT equivalent</U> and if they result to an identity

    C = D,

then you <U>CAN NOT</U> conclude that the original expressions are identical.
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So, working with equivalent transformations, you can transform either side or even both sides - there is  <U>NO obstacles</U>
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.
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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 &nbsp;<U>IS &nbsp;NOT&nbsp; an equivalent transformation</U>:  from &nbsp;&nbsp;{{{a^2}}} = {{{b^2}}}  &nbsp;<U>you can not</U> conclude that &nbsp;a = b.