If \(\alpha\) and \(\beta\) are complementary angles, find the value of the expression: \[ (1 - \sin^2 \alpha)(1 - \cos^2 \alpha)(1 + \cot^2 \beta)(1 + \tan^2 \beta) \]

Q.If \(\alpha\) and \(\beta\) are complementary angles, find the value of the expression: \[ (1 - \sin^2 \alpha)(1 - \cos^2 \alpha)(1 + \cot^2 \beta)(1 + \tan^2 \beta) \]

\((1 - \sin^2 \alpha)(1 - \cos^2 \alpha)(1 + \cot^2 \alpha)(1 + \tan^2 \beta)\) \[ = \cos^2 \alpha \cdot \sin^2 \alpha \cdot \csc^2 \beta \cdot \sec^2 \beta = \cos^2 \alpha \cdot \sin^2 \alpha \cdot \csc^2(90^\circ - \alpha) \cdot \sec^2(90^\circ - \alpha) = \cos^2 \alpha \cdot \sin^2 \alpha \cdot \sec^2 \alpha \cdot \csc^2 \alpha \] \[ = \cos^2 \alpha \cdot \sin^2 \alpha \cdot \frac{1}{\cos^2 \alpha} \cdot \frac{1}{\sin^2 \alpha} = 1 \quad \text{(Answer)} \]