# How do you simplify cos ( 2 Sin^-1 x)?

$\cos \left(2 {\sin}^{-} 1 x\right) = 1 - 2 {x}^{2.}$
Let ${\sin}^{-} 1 x = \theta$, so that, $x = \sin \theta , | x | \le 1 , \theta \in \left[- \frac{\pi}{2} , \frac{\pi}{2}\right] .$
Now, Reqd. Value $= \cos \left(2 {\sin}^{-} 1 x\right) = \cos 2 \theta = 1 - 2 {\sin}^{2} \theta = 1 - 2 {x}^{2.}$