Question

Find the value of x which satisfy equation $\cos \left(2si{n}^{-1}x\right)=\frac{1}{3}$.

• A. $x=\frac{1}{\sqrt{5}}$ and $x=\frac{-1}{\sqrt{5}}$
• B. $x=\frac{1}{\sqrt{3}}$ and $x=\frac{-1}{\sqrt{3}}$
• C. $x=\frac{1}{\sqrt{2}}$ and $x=\frac{-1}{\sqrt{2}}$
• D. None of these

Answer 1

The correct option is B $x=\frac{1}{\sqrt{3}}$ and $x=\frac{-1}{\sqrt{3}}$

Given, $\cos \left(2{\sin }^{-1}x\right)=\frac{1}{3}$

$\Rightarrow \cos \left({\sin }^{-1}\left(2x\sqrt{1-x^2}\right)\right)=\frac{1}{3}$
$\Rightarrow \cos \left({\cos }^{-1}\left(\sqrt{1-{4x}^2+{4x}^4}\right)\right)=\frac{1}{3}\Rightarrow 1-{4x}^2+{4x}^4=\frac{1}{3}$
$\Rightarrow x=\pm \frac{1}{\sqrt{3}}$