Question

If $cos\left(2si{n}^{-1}x\right)=\frac{1}{9}$, then x=

• A. $Only\frac{2}{3}$
• B. $Only\frac{-2}{3}$
• C. $\frac{2}{3},\frac{-2}{3}$
• D. $Neither\frac{2}{3}nor\frac{-2}{3}$

Answer 1

The correct option is C $\frac{2}{3},\frac{-2}{3}$

$cos\left(2si{n}^{-1}x\right)=\frac{1}{9}$
$\Rightarrow cos\left(si{n}^{-1}2x\sqrt{1-x^2}\right)=\frac{1}{9}$
$\Rightarrow cos\left(co{s}^{-1}\sqrt{1-4x^2+4x^4}\right)=\frac{1}{9}$
$\Rightarrow 1-2x^2=\frac{1}{9}\Rightarrow 2x^2=1-\frac{1}{9}=\frac{8}{9}$
$\Rightarrow x^2=\frac{4}{9}\Rightarrow x=\pm \frac{2}{3}$.