Step-by-step Solution:

1. Pairing Tangent Values: Notice that \( \tan(90^\circ - \theta) = \cot \theta \). Therefore, we can pair the terms as follows: \[ \tan 1^\circ \tan 89^\circ = \tan 1^\circ \cot 1^\circ = \tan 1^\circ \cdot \frac{1}{\tan 1^\circ} = 1 \] \[ \tan 2^\circ \tan 88^\circ = \tan 2^\circ \cot 2^\circ = \tan 2^\circ \cdot \frac{1}{\tan 2^\circ} = 1 \] \[ \vdots \] \[ \tan 44^\circ \tan 46^\circ = \tan 44^\circ \cot 44^\circ = \tan 44^\circ \cdot \frac{1}{\tan 44^\circ} = 1 \] 2. Middle Term: The middle term is \( \tan 45^\circ \), which is: \[ \tan 45^\circ = 1 \] 3. Calculate the Product: Since each pair multiplies to 1 and the middle term is also 1, the entire product is: \[ \tan 1^\circ \tan 2^\circ \tan 3^\circ \ldots \tan 89^\circ = 1 \] Therefore, the value of the product is: \[ \boxed{1} \]