YES

(VAR x0 x1 x2 a b)
(RULES
  divide(x0,divide(x1,x2)) -> divide(divide(x0,inverse(x2)),x1)
  inverse(divide(x1,x0)) -> divide(x0,x1)
  divide(divide(x0,x2),inverse(x2)) -> x0
  divide(divide(x0,inverse(x1)),x1) -> x0
  inverse(inverse(x1)) -> x1
  divide(x0,identity()) -> x0
  inverse(identity()) -> identity()
  divide(a,a) -> identity()
  multiply(a,b) -> divide(a,inverse(b))
  divide(identity(),a) -> inverse(a)
)

(COMMENT
Termination is shown by EKBO with interpretations on natural numbers

  divide_A(x1,x2) = x1 + x2 + 1
  identity_A = 0
  multiply_A(x1,x2) = x1 + x2 + 1
  inverse_A(x1) = x1
  divide#_A(x1,x2) = x2
  identity#_A = 0
  multiply#_A(x1,x2) = x1 + x2

weights

  w0 = 2
  w(divide) = 0
  w(identity) = 2
  w(multiply) = 0
  w(inverse) = 0

and precedence:

inverse > multiply > divide > identity
)