Prove that `2tan^(-1)1/x=sin^(-1)((2x)/(x^(2)+1))`

Explanation

To prove that \( 2\tan^{-1}\left(\frac{1}{x}\right) = \sin^{-1}\left(\frac{2x}{x^2 + 1}\right) \), we will start with the left-hand side (LHS) and manipulate it to show that it equals the right-hand side (RHS).

### Step 1: Define \( \theta \)
Let
\[ \theta = \tan^{-1}\left(\frac{1}{x}\right) \]
This implies that
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