# How do you prove sin^-1(-x)=-sin^-1x?

Oct 12, 2016

see below

#### Explanation:

${\sin}^{-} 1 \left(- x\right) = - {\sin}^{-} 1 x$

Let $y = - {\sin}^{-} 1 x$

$- y = {\sin}^{-} 1 x$

$\sin \left(- y\right) = x$ by the inverse property

$- \sin y = x$ by the odd function property $\sin \left(- x\right) = - \sin x$

$\sin y = - x$

$y = {\sin}^{-} 1 \left(- x\right)$

$\therefore - {\sin}^{-} 1 x = {\sin}^{-} 1 \left(- x\right)$