# How to expand all portions of an expression?

Consider a toy expression:

expr = Sin[x (a + b)]/(y (z + c))


I would like to expand all parentheses to get something like:

Sin[a x + b x]/(c y + y z)


I have been able to expand the argument of the Sin term using:

MapAll[Expand,expr]
(* Out[] = Sin[a x + b x]/(y (c + z)) *)


However, this leaves the denominator unexpanded. Is there a way to expand the entire expression?

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ExpandAll will do this. –  Daniel Lichtblau May 29 '12 at 21:24
@DanielLichtblau Yeah, I found that function like 2 seconds after I posted the question. Any idea why regular Expand won't do it? –  Eli Lansey May 29 '12 at 21:25
Best I recall it was only meant to handle polynomial-like things. Before my time, really. –  Daniel Lichtblau May 29 '12 at 21:44
@DanielLichtblau "Before my time" whaa? Aren't you a dinosaur at WRI? –  rm -rf May 29 '12 at 23:08
No, just a mastodon. Mastodanny, actually, but they are cousin species. –  Daniel Lichtblau May 29 '12 at 23:09

I just found the ExpandDenominator function which, as the name implies, expands the denominator of the expression. Then, Expand and ExpandDenominator can be cobbled together using:

ExpandDenominator //@ Expand //@ expr


(Thanks Mr. Wizard for the improved cleanliness.) However, there is also a related built-in function that does this all in one shot:

ExpandAll[expr]
(* Out[] = Sin[a x + b x]/(c y + y z) *)

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The first line looks better (to my eye) as: ExpandDenominator //@ Expand //@ expr –  Mr.Wizard May 30 '12 at 2:18
@Mr.Wizard Good call. Edited the post to clean up that expression. –  Eli Lansey May 31 '12 at 13:36

It seems that an expression like 1/((a + b) x) is totally deceiving. It doesn't remain unevaluated, it evaluates to something, but that other something is formatted in the same way as the original.

1/((a + b) x) // FullForm // HoldForm


Times[1, Power[Times[Plus[a, b], x], -1]]

1/((a + b) x) // FullForm


Times[Power[Plus[a, b], -1], Power[x, -1]]

So, as you can see, it evaluates to something that should look like 1/(a+b) 1/x That's why Expand never gets mapped to a polynomial denominator.

Try, for example

Expand//@Unevaluated[Sin[x (a + b)]/(y (z + c))]


Sin[a x + b x]/(c y + y z)

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