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  4. The set of values of x satisfying the inequation tan 2( sin -1 x) >1, is

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Q. The set of values of $x$ satisfying the inequation $\tan ^{2}\left(\sin ^{-1} x\right) >1$, is

Inverse Trigonometric Functions

Solution:

We have,
$\Rightarrow \tan ^{2}\left(\sin ^{-1} x\right)>1 $
$\Rightarrow \tan ^{2}\left(\sin ^{-1} x\right)-1>0 $
$\Rightarrow \tan \left(\sin ^{-1} x\right)<-1 $
or, $\tan \left(\sin ^{-1} x\right)>1 $
$\Rightarrow -\infty<\tan \left(\sin ^{-1} x\right)<-1 $
or, $ 1<\tan \left(\sin ^{-1} x\right)<\infty $
$\Rightarrow -\frac{\pi}{2}<\sin ^{-1} x<-\frac{\pi}{4} $
or, $ \frac{\pi}{4}<\sin ^{-1} x<\frac{\pi}{2} $
$\Rightarrow x \in\left(-1,-\frac{1}{\sqrt{2}}\right) $
or, $x \in\left(\frac{1}{\sqrt{2}}, 1\right) $
$\Rightarrow x \in(-1,-1)-\left[-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right]$