First slide

Properties of ITF

Question

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

Moderate

A

$[- 1, 1]$
B

$[- \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}]$
C

$(- 1, 1) - [- \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}]$
D

$[- 1, 1] - (- \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$

Solution

We have,
$\begin{array}{l} \tan^{2} (\sin^{- 1} x) > 1 \\ \Rightarrow \tan^{2} (\sin^{- 1} x) - 1 > 0 \\ \Rightarrow \tan (\sin^{- 1} x) < - 1 or, \tan (\sin^{- 1} x) > 1 \\ \Rightarrow - \infty < \tan (\sin^{- 1} x) < - 1 or, 1 < \tan (\sin^{- 1} x) < \infty \\ \Rightarrow - \frac{π}{2} < \sin^{- 1} x < - \frac{π}{4} or, \frac{π}{4} < \sin^{- 1} x < \frac{π}{2} \\ \Rightarrow x \in (- 1, - \frac{1}{\sqrt{2}}) or, x \in (\frac{1}{\sqrt{2}}, 1) \\ \Rightarrow x \in (- 1, - 1) - [- \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}] \end{array}$

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