The minimum value of $x$ which satisfies the inequality ${\left(\mathrm{s}\mathrm{i}{\mathrm{n}}^{-1}x\right)}^2\ge {\left(\mathrm{c}\mathrm{o}{\mathrm{s}}^{-1}x\right)}^2$ is

• Correct answer
  A

  $\frac{1}{\sqrt{2}}$
• B

  $\frac{1}{2}$
• C

  $\frac{\sqrt{3}}{2}$
• D

  $\frac{1}{\sqrt{3}}$

Solution

${\left(\mathrm{s}\mathrm{i}{\mathrm{n}}^{-1}x\right)}^2-{\left(\mathrm{c}\mathrm{o}{\mathrm{s}}^{-1}x\right)}^2\ge 0$
$\left(\mathrm{s}\mathrm{i}{\mathrm{n}}^{-1}x+\mathrm{c}\mathrm{o}{\mathrm{s}}^{-1}x\right)\left(\mathrm{s}\mathrm{i}{\mathrm{n}}^{-1}x-\mathrm{c}\mathrm{o}{\mathrm{s}}^{-1}x\right)\ge 0$
$\Rightarrow \mathrm{s}\mathrm{i}{\mathrm{n}}^{-1}x\ge \mathrm{c}\mathrm{o}{\mathrm{s}}^{-1}x$
Clearly from graph
$x\in \left[\frac{1}{\sqrt{2}},1\right]$

Hence, the correct answer is option: 'A'.