Question

How do you simplify $\cos \left(si{n}^{-1}x\right)$ ?

Answer 1

$\pm \sqrt{1-x^2}$
Explanation:
$cos\left(si{n}^{-1}x\right)$
Let,
$si{n}^{-1}x=\theta$
$\Rightarrow sin\theta =x$
$\Rightarrow si{n}^2\theta =x^2$
$\Rightarrow 1-co{s}^2\theta =x^2$
$\Rightarrow co{s}^2\theta =1-x^2$
$\Rightarrow cos\theta =\pm \sqrt{1-x^2}$
$\Rightarrow \theta =co{s}^{-1}\pm \sqrt{1-x^2}$
Putting this,
$cos(co{s}^{-1}\pm \sqrt{1-x^2})$
$=\pm \sqrt{1-x^2}$
But $si{n}^{-1}x$ is, definition, in $[-\frac{\pi }{2},\frac{\pi }{2}]$ so $cos\left(si{n}^{-1}x\right)\ge 0$
so $cos\left(si{n}^{-1}x\right)=\sqrt{1-x^2}$