If (2 ^ - 1 x) = 1 9 , then x = - Vidyadip

MCQ

If $\cos (2{\sin ^{ - 1}}x) = \frac{1}{9},$ then $x = $

A
Only $ \frac{2}{3}$
B
Only$ \frac{-2}{3}$
✓
$\frac{2}{3}, \frac{-2}{3}$
D
Neither $\frac{2}{3}$ nor $\frac{-2}{3}$

✓

Answer

Correct option: C.

$\frac{2}{3}, \frac{-2}{3}$

c
(c) $\cos (2{\sin ^{ - 1}}x) = \frac{1}{9}$

$ \Rightarrow \cos ({\sin ^{ - 1}}2x\sqrt {1 - {x^2}} ) = \frac{1}{9}$

==>$\cos ({\cos ^{ - 1}}\sqrt {1 - 4{x^2} + 4{x^4}} ) = \frac{1}{9}$

==> $1 - 2{x^2} = \frac{1}{9} \Rightarrow 2{x^2} = 1 - \frac{1}{9} = \frac{8}{9}$

==> ${x^2} = \frac{4}{9} \Rightarrow x = \pm \frac{2}{3}$.

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