We know that 

sin^-1 x = cosec^-1 1/x ...............1

Now 1/sinx = (sinx)^-1

=> 1/sinx = (1/cosecx)^-1

=> 1/sinx = cosecx .................2

From equation 1 and 2, we get

 sin^-1 x ≠ 1/sinx

So, sin^-1 x can not be written as 1/sinx