Question

If x < 0, then find the value of
$2\left({\tan }^{-1}\frac{1}{x}+{\tan }^{-1}x\right)$

• A. $\pi$
• B. $-\pi$
• C. $\frac{\pi }{2}$
• D. $-\frac{\pi }{2}$

Answer 1

The correct option is B $-\pi$

We know that
${\tan }^{-1}\frac{1}{x}=(\begin{array}{cc}co{t}^{-1}x,& x>0\\ -\pi +{\cot }^{-1}x,& x<0\end{array}\mathrm{Therefore},2({\tan }^{-1}\frac{1}{x}+{\tan }^{-1}x)=2(-\pi +{\cot }^{-1}x+{\tan }^{-1}x)=2(-\pi +\frac{\pi }{2}\left)\right(\because {\tan }^{-1}x+co{t}^{-1}x=\frac{\pi }{2},x\in R]=-\pi$