\[ \tan \alpha = \cot \beta \Rightarrow \tan \alpha = \tan(90^\circ - \beta) \Rightarrow \alpha = 90^\circ - \beta \Rightarrow \alpha + \beta = 90^\circ \] \[ \therefore \cos(\alpha + \beta) = \cos 90^\circ = 0 \quad \text{(Answer)} \]
\[ \tan \alpha = \cot \beta \Rightarrow \tan \alpha = \tan(90^\circ - \beta) \Rightarrow \alpha = 90^\circ - \beta \Rightarrow \alpha + \beta = 90^\circ \] \[ \therefore \cos(\alpha + \beta) = \cos 90^\circ = 0 \quad \text{(Answer)} \]