I remember someone once asked about checking the equality of equations like,
Sqrt(5-2*Sqrt(6))=Sqrt(3)-Sqrt(2) in Mathemaica.
"FullSimplify[]" is a good solution.
After reading Edward R. Scheinerman's article "When Close Enough Is Close Enough" published on American Mathematical Monthly June-July 2000, I know how to check equality "manually" if FullSimplify happens to fail.
I implement a function according to Scheinerman's theory.
I recommend you read his original paper if possible. Or ask me to send you a Notebook for details, if you are interested in this subject. (I don't know if it is a good practice to send a message with attachment file, otherwise I would have attached my notebook with this email.)
By the way, I don't know how FullSimplify[] checks equality. Hope some experts may divulge this secret.