Simplification with assumptions

Question

I'm trying to make Mathematica do some simplifications for me. I have the expression

((1 + s[1]) (-1 + s[2]) )/(-1 + s[1] s[2])

and want to simplify this with the assumption that

s[1]^2==1

(it should simplify to 1+s[1], as can be verified manually)

I obviously tried

Simplify[((1 + s[1]) (-1 + s[2]) )/(-1 + s[1] s[2]), s[1]^2 == 1]

to no avail. Is there any way I can persuade Mathematica to do this for me?

Edit:

Here's the manual simplification:

$$ \frac{(1 + s[1]) (-1 + s[2])}{-1 + s[1] s[2]}=\frac{(1 + s[1])(-1+s[1]^{-1})(-1+s[1]s[2])+(-1+s[1]^{-1})+(-1+s[1]s[2]) }{-1 + s[1] s[2]}=\frac{(1 + s[1])((-1+s[1])(-1+s[1]s[2])+(-1+s[1])+(-1+s[1]s[2]))}{-1 + s[1] s[2]}=\frac{(1 + s[1])((-1+s[1])(-1+s[1]s[2])+(-1+s[1])+(-1+s[1]s[2]))}{-1 + s[1] s[2]}=\frac{(1 + s[1])(-1+s[1])((-1+s[1]s[2])+1)+(1+s[1])(-1+s[1]s[2])}{-1 + s[1] s[2]}=\frac{0((-1+s[1]s[2])+1)+(1+s[1])(-1+s[1]s[2])}{-1 + s[1] s[2]}=1+s[1] $$

Answer 1

Simplify has built-in support for the Assumptions option. Thus, for example,

Simplify[Sqrt[x^2]]

will return the same argument, since the real part of x could be negative, then it can simplify it further:

Simplify[Sqrt[x^2], Assumptions -> x > 0]

returns x.

---

In your case, though, there is no ready simplification because there's no real need to call s[1]^2. On the other hand, one can see that your assumption severely restricts the values s[1] can take. I would advise you to see this in the light of "s[1] must satisfy this given equation": you can then Solve it and see what happens to your expression when you substitute in the values. Thus,

((1 + s[1]) (-1 + s[2]))/(-1 + s[1] s[2]) /. Solve[s[1]^2 == 1, s[1]]

returns {0,2}, which is what I believe you were expecting. If your expression gets more complicated, you may need to bring in a further Simplify step enveloping the substitution.

Answer 2

   (((1 + s[1]) (-1 + s[2]))/(-1 + s[1] s[2]) // Expand) /. 
  s[1] -> 1 // Simplify

(*   2    *)

    (((1 + s[1]) (-1 + s[2]))/(-1 + s[1] s[2]) // Expand) /. 
  s[1] -> -1 // Simplify

(*  0   *)

Have fun.