The angles A + B and A − B are both acute and tan(A + B)·tan(A − B) = 1. Then A equals:

For acute angles, tanX·tanY = 1 means tanY = cotX, so X and Y are complementary. Hence $(A+B)+(A-B)=90$°, which is $2A=90$° and $A=45$°. $B$ itself is not fixed — for instance $A+B=55$° with $A-B=35$° works, giving $B=10$°. Reporting the sum leaves 90°, halving once more gives 22.5°, and 60° would need the two angles to total 120°, where the product of the tangents is not 1.