Simplifying the body of a pure function

fun = Function[{x}, Cos[x] Cot[g]/Sqrt[-Cos[a]^2 Cot[g]^2 + Sin[a]^2]];

fun[[2]] = 
  FullSimplify[fun[[2]] /. a -> ArcSin[k Cos[g]], 0 < g < Pi/2];

fun