Question:

Step 1: Recall the identity for inverse tangent when x > 0. For positive values of $x$, we have the fundamental identity: $$\tan^{-1}\left(\frac{1}{x}\right) = \frac{\pi}{2} - \tan^{-1}(x)$$ Step 2: Substitute the identity into the given equation. The given equation is: $$\tan^{-1}\left(\frac{1}{x}\right) = \pi + \tan^{-1}(x)$$ Substituting the identity from Step 1: $$\frac{\pi}{2} - \tan^{-1}(x) = \pi + \tan^{-1}(x)$$ Step 3: Rearrange to isolate $\tan^{-1}(x)$. Moving all terms with $\tan^{-1}(x)$ to one side: $$\frac{\pi}{2} - \pi = \tan^{-1}(x) + \tan^{-1}(x)$$ $$-\frac{\pi}{2} = 2\tan^{-1}(x)$$ Step 4: Solve for $\tan^{-1}(x)$. Dividing both sides by 2: $$\tan^{-1}(x) = -\frac{\pi}{4}$$ Step 5: Find the value of x. Taking the tangent of both sides: $$x = \tan\left(-\frac{\pi}{4}\right) = -1$$ Step 6: Check if the solution satisfies the given constraint. We obtained $x = -1$, but the problem requires $0 < x < 1$. Since $x = -1$ does not satisfy the constraint $0 < x < 1$, this solution is invalid. **Conclusion:** There are no values of $x$ in the interval $(0, 1)$ that satisfy the given equation. **Answer: Option 1 (0)**