Step-by-step Solution:

The given expression can be written as: \[ 2 \tan^{-1} \left[ \csc \left( \csc^{-1} \left( \frac{\sqrt{1 + x^2}}{x} \right) \right) \right] - \tan \left( \tan^{-1} \left( \frac{1}{x} \right) \right) \] \[ = 2 \tan^{-1} \left[ \frac{\sqrt{1 + x^2}}{x} - \frac{1}{x} \right] = 2 \tan^{-1} \left[ \frac{\sqrt{1 + x^2} - 1}{x} \right] \] Put \( x = \tan \theta \), then: \[ = 2 \tan^{-1} \left[ \frac{\sec \theta - 1}{\tan \theta} \right] = 2 \tan^{-1} \left( \frac{1 - \cos \theta}{\sin \theta} \right) \] \[ = 2 \tan^{-1} \left( \frac{2 \sin^2 \frac{\theta}{2}}{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}} \right) = 2 \tan^{-1} \left( \tan \frac{\theta}{2} \right) \] \[ = \theta = \tan^{-1} x \]