Question

If $x$ satisfies the inequality ${\left({\tan }^{-1}x\right)}^2+3\left({\tan }^{-1}x\right)-4>0,$ then the complete set of values of $x$ is

• A. $\left(-\tan 4,\frac{\pi }{4}\right)$
• B. $(-\infty ,\tan 4)\cup \left(\frac{\pi }{4},\infty \right)$
• C. $(\tan 1,\infty )$
• D. $(\tan 4,\tan 1)$

Answer 1

Answer: (C)

Let, ${\tan }^{-1}x=t$
the given inequality becomes $t^2+3t-4>0\Rightarrow (t+4)(t-1)>0$
$\therefore t<-4$ or $t>1$
${\tan }^{-1}x<-4$ (not possible) or ${\tan }^{-1}x>1$
$\Rightarrow x>\tan 1$