Step-by-step Solution:

\[ \text{Given} \] \[ A + B = 45^\circ \] \[ \tan (A + B) = \tan 45^\circ \] \[ \frac{\tan A + \tan B}{1 - \tan A \tan B} = 1 \] \[ \tan A + \tan B = 1 - \tan A \tan B \] \[ \tan A + \tan B + \tan A \tan B = 1 \] \[ \text{Adding } 1 \text{ on both sides} \] \[ 1 + \tan A + \tan B + \tan A \tan B = 2 \] \[ 1(1 + \tan A) + \tan B (1 + \tan A) = 2 \] \[ (1 + \tan A)(1 + \tan B) = 2. \]