First slide

Properties of ITF

Question

The set of values of $x$ for which $\tan^{- 1} \frac{x}{\sqrt{1 - x^{2}}} = \sin^{- 1} x$ holds, is

Moderate

A

R
B

$(- 1, 1)$
C

$[0, 1]$
D

$[- 1, 0]$

Solution

We observe that RHS is defined for all $x \in [- 1, 1]$
whereas LHS is meaningful for $- 1 < x < 1$
Also, for $x \in (- 1, 1)$ , we have
$\begin{aligned} \tan^{- 1} (\frac{x}{\sqrt{1 - x^{2}}}) \end{aligned}$
$= \tan^{- 1} (\frac{\sin θ}{\cos θ})$ , where $x = \sin θ$
$= θ = \sin^{- 1} x$

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