If A + B = 45° and neither angle is 90°, then (1 + tanA)(1 + tanB) equals:

tan$(A+B)=1$ gives (tanA + tanB)/(1 − tanA·tanB) = 1, so tanA + tanB = 1 − tanA·tanB. Expanding the product: 1 + (tanA + tanB) + tanA·tanB = 1 + 1 − tanA·tanB + tanA·tanB = 2. Check with $A=10$°, $B=35$°: $1.1763\times 1.7002=2.000$.