I know this isn't true:

$\displaystyle

\sin ^{ - 1} x \ne \frac{1}

{{\sin x}}

$

Yet, does the same rules apply to inverse functions regarding the reciprocals? For example:

$\displaystyle

\sin x = \frac{1}

{{\csc x}}

$

Still my main question remains; Is this equality true? Can I apply the same known trigonometric reciprocals with the inverse trig functions?

$\displaystyle

\sin ^{ - 1} x = \frac{1}

{{\csc ^{ - 1} x}}

$

Thank you.