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

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

Inverse Trigonometric Functions

Solution:

$\tan ^2\left(\sin ^{-1} x\right)>1$
either $\tan \left(\sin ^{-1} x\right) >1 \Rightarrow \sin ^{-1} x >\tan ^{-1} 1$
$\Rightarrow \sin ^{-1} x >\sin ^{-1} \frac{1}{\sqrt{2}}$
$\Rightarrow x>\frac{1}{\sqrt{2}} \Rightarrow \frac{1}{\sqrt{2}}< x< 1$
or $\tan \left(\sin ^{-1} x\right)< -1 \Rightarrow \sin ^{-1} x< \tan ^{-1}(-1)$
$\Rightarrow \sin ^{-1} x< \sin ^{-1}\left(-\frac{1}{\sqrt{2}}\right) \Rightarrow-1< x< -\frac{1}{\sqrt{2}}$
so $ x \in(-1,1)-\left[\frac{-\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right]$