Question

The range of values of ${}^{\prime }{x}^{\prime }$ satisfying the inequation, ${\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)<\frac{\pi }{3}$ are

• A. $xϵ[-\frac{1}{3},\frac{1}{\sqrt{3}}]$
• B. $xϵ(-\frac{1}{3},\frac{1}{\sqrt{3}})$
• C. $xϵ(0,\frac{1}{\sqrt{3}})$
• D. $noneofthese$

Answer 1

The correct option is B $xϵ(-\frac{1}{3},\frac{1}{\sqrt{3}})$

$co{s}^{-1}\left(\frac{1-x^2}{1+x^2}\right)<\frac{\pi }{3}$
Taking $cosine$ on both sides, we get
$\frac{1-x^2}{1+x^2}>\frac{1}{2}$

$\Rightarrow 2-2x^2>1+x^2$

$\Rightarrow 1>3x^2$

$\Rightarrow x^2<\frac{1}{3}$

$\Rightarrow |x|<\frac{1}{\sqrt{3}}$

$\Rightarrow -\frac{1}{\sqrt{3}}<x<\frac{1}{\sqrt{3}}$