Even though it's correct to calculate tangent in order to check the statement, shouldn't Wolfram do it after expanding the brackets?
Mathematica syntax uses Tan[z], ArcTan[z].
There's a difference in Mathematica between Equal or == and SameQ or ===. The triple-equal relation tests whether the expressions are identical. It does not try any transformations to see whether they are mathematically equivalent. Apparently W|A interprets the triple-equal in the WL way. W|A seems to work ok with = or == (that is, it returns "True"). I suppose the W|A developers felt that === is not standard notation except in Mathematica and assigned it the SameQ interpretation.
Just FYI, you can choose "Wolfram | Alpha" in your groups