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 $t^{2}+3 t-4>0 \Rightarrow (t+4)(t-1)>0$
$\therefore t<-4$ or $t>1$
$\tan ^{-1} x<-4$ (not possible) or $\tan ^{-1} x>1$
$\Rightarrow x>\tan 1$