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,

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 < l.
Also, for $x \in (- 1, 1)$ , we have
$\begin{array}{l} \tan^{- 1} (\frac{x}{\sqrt{1 - x^{2}}}) \\ = \tan^{- 1} (\frac{\sin θ}{\cos θ}), where x = \sin θ \\ = θ = \sin^{- 1} x \end{array}$

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