       Re: A Problem with Simplify

• To: mathgroup at smc.vnet.net
• Subject: [mg87497] Re: A Problem with Simplify
• From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
• Date: Fri, 11 Apr 2008 05:57:47 -0400 (EDT)
• Organization: The Open University, Milton Keynes, UK
• References: <ftkb7f\$a9m\$1@smc.vnet.net> <ftmts8\$4l5\$1@smc.vnet.net>

```Alexey Popkov wrote:

<snip>

> Recently I also have found another strange bug in Simplfy (not so
> dangerous):
> FullSimplify[Sin[x]*Cos[x]]
> gives Sin[x]*Cos[x] instead Sin[2*x]/2. The workaround is
> FullSimplify[2*Sin[x]*Cos[x]]/2.

<snip>

This is not a bug, indeed. As a rule of thumb, *FullSimplify* tries to
minimize the leaf count of the expression.

For the case above, the original expression sin(x)cos(x) has a leaf
count of 5 whereas the trigonometric identity sin(2x)/2 has a leaf count
of 8. Therefore, *FullSimplify* returns the original form sin(x)cos(x)
since it is deemed as simpler than sin(2x)/2 w.r.t. the lead count of
each expression.

Note that, in general, you can modify/tweak this default behavior to
dealing with trigonometric expressions (especially when looking for some
trigonometric identities) it is usually better and less cumbersome to
use  *TrigReduce* and the like (*TrigExpand*, *TrigFactor*, etc.).

FullSimplify[Sin[x]*Cos[x]]      (* returns  Cos[x] Sin[x]  *)
% // LeafCount                   (* returns  5              *)
TrigReduce[Sin[x]*Cos[x]]        (* returns  1/2 Sin[2 x]   *)
% // LeafCount                   (* returns  8              *)
LeafCount[Cosh[x] - Sinh[x]]     (* returns  7              *)
FullSimplify[Cosh[x] - Sinh[x]]  (* returns  E^-x           *)
% // LeafCount                   (* returns  5              *)

Best regards,
-- Jean-Marc

```

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