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Find the values of $A$ and $B$ ($0 \leq A < 90^{\circ}, 0 \leq B < 90^{\circ}$), if $\tan(A + B) = 1$ and $\tan(A - B) = \frac{1}{\sqrt{3}}$.
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Solution: (a) $A + B = 45^{\circ}$
$A - B = 30^{\circ}$
Solving and getting $A = 37.5^{\circ}$ and $B = 7.5^{\circ}$
$A - B = 30^{\circ}$
Solving and getting $A = 37.5^{\circ}$ and $B = 7.5^{\circ}$