If `alpha = sin^(-1) (cos(sin^-1) x)) and beta = cos^(-1) (sin (cos^(-1) x))`, then find `tan alpha. tan beta.`

Explanation

To solve the problem, we need to find the value of \( \tan \alpha \cdot \tan \beta \) where \( \alpha = \sin^{-1}(\cos(\sin^{-1} x)) \) and \( \beta = \cos^{-1}(\sin(\cos^{-1} x)) \).

### Step-by-step Solution:

1. **Understanding the relationship between \( \alpha \) and \( \beta \)**:
We know that \( \cos^{-1} x + \sin^{-1} x = \frac{\pi}{2} \). Therefore, we can express \( \beta \) in terms of \( \alpha \):
\[
\beta = \cos^{-1}(\sin(\cos^{-1} x)) = \frac{\pi}{2} - \sin^{-1} x
...