Answer
$\csc^{-1}{\frac{1}{2}}$ is undefined and $\csc^{-1}{{2}}$ is defined
Work Step by Step
a) To determine if $\csc^{-1}{\frac{1}{2}}$ is defined, we note the following:
$\csc{x}={\frac{1}{2}}$
${\frac{1}{\sin{x}}}={\frac{1}{2}}$
${\sin{x}}=2$
But that value is outside the range of sin, thus $\csc^{-1}{\frac{1}{2}}$ is undefined.
b) To determine if $\csc^{-1}{{2}}$ is defined, we note the following:
let $\csc^{-1}{{2}}=x$
$\csc{x}={{2}}$
${{\sin{x}}}={\frac{1}{2}}$
Which is inside the range of sin, thus $\csc^{-1}{{2}}$ is defined.
