Question:
$\tan^{-1}(\frac{1001}{999}) - \tan^{-1}(\frac{2}{2000}) = $ ________.
$\tan^{-1}(\frac{1001}{999}) - \tan^{-1}(\frac{2}{2000}) = $ ________.
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Simplify the argument before calculating the inverse tangent.
Updated On: Jun 26, 2026
- $\frac{\pi}{3}$
- $\pi$
- 1
- $\frac{\pi}{6}$
- $\frac{\pi}{4}$
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The Correct Option is
Solution and Explanation
Step 1: Concept
Apply the formula $\tan^{-1} A - \tan^{-1} B = \tan^{-1}(\frac{A-B}{1+AB})$.
Step 2: Meaning
Let $A = \frac{1001}{999}$ and $B = \frac{2}{2000} = \frac{1}{1000}$.
Step 3: Analysis
$\frac{A-B}{1+AB} = \frac{\frac{1001}{999} - \frac{1}{1000}}{1 + \frac{1001}{999000}} = \frac{\frac{1001000 - 999}{999000}}{\frac{999000 + 1001}{999000}} = \frac{1000001}{1000001} = 1$.
Step 4: Conclusion
$\tan^{-1}(1) = \frac{\pi}{4}$. Final Answer: (E)
Apply the formula $\tan^{-1} A - \tan^{-1} B = \tan^{-1}(\frac{A-B}{1+AB})$.
Step 2: Meaning
Let $A = \frac{1001}{999}$ and $B = \frac{2}{2000} = \frac{1}{1000}$.
Step 3: Analysis
$\frac{A-B}{1+AB} = \frac{\frac{1001}{999} - \frac{1}{1000}}{1 + \frac{1001}{999000}} = \frac{\frac{1001000 - 999}{999000}}{\frac{999000 + 1001}{999000}} = \frac{1000001}{1000001} = 1$.
Step 4: Conclusion
$\tan^{-1}(1) = \frac{\pi}{4}$. Final Answer: (E)
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