TITLE: Is this trig step correct?
QUESTION [5 upvotes]: $\sin^{-1}(-\sin(x))$ = $-\sin^{-1}(\sin(x))$
Can the minus be taken out like this?

REPLY [4 votes]: Using the definition of principal value of inverse sine function,
we can always find an integer $n$ such that $\displaystyle x=n\pi+(-1)^n y$ with  $\displaystyle -\frac\pi2\le y\le\frac\pi2$
$\displaystyle\implies-\frac\pi2\le-y\le\frac\pi2 $ 
$\displaystyle\implies\sin x=\sin(n\pi+(-1)^n y)=\sin y $
$\displaystyle\implies-\sin x=-\sin y=\sin(-y)$
$\displaystyle\implies\sin^{-1}(\sin(-y))=-y$ as $\displaystyle-\frac\pi2\le-y\le\frac\pi2 $
Similarly, $\displaystyle\sin^{-1}(\sin x)=\sin^{-1}(\sin y)=y$ as $\displaystyle-\frac\pi2\le y\le\frac\pi2 $