Question

Assertion :If $x<0,{\tan }^{-1}x+{\tan }^{-1}\frac{1}{x}=\frac{\pi }{2}$ Reason: ${\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}\forall x\in R$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct.

Answer 1

The correct option is D Assertion is incorrect but Reason is correct.

${\tan }^{-1}\left(x\right)+{\tan }^{-1}\left(\frac{1}{x}\right)$ $={\tan }^{-1}\left(x\right)+{\cot }^{-1}\left(x\right)$

Now consider the following

$\tan \left(\theta \right)=\cot ({90}^0-\theta )$.

This relation is true for all real values of $\theta$.

Therefore

${\tan }^{-1}\left(x\right)+{\cot }^{-1}\left(x\right)=\frac{\pi }{2}$ is true for all $xϵR$. ...(domain and range of ${\tan }^{-1}\left(x\right),{\cot }^{-1}\left(x\right)$ is $(-\infty ,\infty )$.

Hence assertion is false, whereas reason is true.