UPSC NDA I 2021 Mathematics Solved Paper

Concept:The identity $\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2}$ is a fundamental property of inverse trigonometric functions.Explanation:For any real number $x$, the inverse cotangent function can be expressed as $\cot^{-1}x = \frac{\pi}{2} - \tan^{-1}x$.This relation holds for all $x \in \mathbb{R}$ because the sum of an angle and its complement is $\frac{\pi}{2}$ in the principal value ranges of $\tan^{-1}x$ and $\cot^{-1}x$.Adding $\tan^{-1}x$ to both sides gives $\tan^{-1}x + \cot^{-1}x = \tan^{-1}x + \left(\frac{\pi}{2} - \tan^{-1}x\right) = \frac{\pi}{2}$.No restrictions on $x$ are needed; the equality is valid for every real number.Thus the given equation holds for all $x \in \mathbb{R}$.Answer:Option A: $x \in \mathbb{R}$