If g is the inverse of function f and f'(x) = sin x, then g'(x) =

$\frac{1}{\sin \{g(x)\}}$

Since g is the inverse of function f. Therefore,

fog(x) = x  for all  x

$\Rightarrow \frac{d}{d x}(fog(x))=\frac{d}{d x}(x)$  for all  x

$\Rightarrow \frac{d}{d g(x)}\{f(g(x))\} . g'(x)=1$  for all  x

$\Rightarrow \sin \{g(x)\} g'(x)=1$  for all  x

$\Rightarrow g'(x)=\frac{1}{\sin \{g(x)\}}$  for all  x