First slide

Properties of ITF

Question

If $x \in [- 1, 1]$ then $\sin^{- 1} (\frac{2 x}{1 + x^{2}})$ equals

Moderate

A

$2 \tan^{- 1} x$
B

$π - 2 \tan^{- 1} x$
C

$- π - 2 \tan^{- 1} x$
D

none of these

Solution

Let $\tan^{- 1} x = θ .$ Then $x = \tan θ .$
Now,
$- 1 \leq x \leq 1 \Rightarrow - 1 \leq \tan θ \leq 1 \Rightarrow - \frac{π}{4} \leq θ \leq \frac{π}{4} \Rightarrow - \frac{π}{2} \leq 2 θ \leq \frac{π}{2}$
$\begin{aligned} ∴ \sin^{- 1} (\frac{2 x}{1 + x^{2}}) \\ = \sin^{- 1} (\sin 2 θ) \end{aligned}$
$= 2 θ [∵ - \frac{π}{4} \leq θ \leq \frac{π}{4} \Rightarrow - \frac{π}{2} \leq 2 θ \leq \frac{π}{2}]$
$= 2 \tan^{- 1} x$

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