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how to simplify a complicated equation?

• To: mathgroup at smc.vnet.net
• Subject: [mg43529] how to simplify a complicated equation?
• From: timreh719 at yahoo.com.tw (bryan)
• Date: Fri, 19 Sep 2003 03:41:52 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

Dear All,
  I'm dealing with the problem about the coordinate rotation matrix
around an arbitrary axis; and I have a correct answer. How can I use
some constraints to reduce or simplify an equation to desired form ?
For example, here

In[1]:  correct={{a^2 (1-Cos[t])+Cos[t],a b (1-Cos[t])+c Sin[t],a c
(1-Cos[t])-b Sin[t]},{a b (1-Cos[t])-c Sin[t],b^2 (1-Cos[t])+Cos[t],b
c (1-Cos[t])+a Sin[t]},{a c (1-Cos[t])+b Sin[t],b c (1-Cos[t])-a
Sin[t],c^2 (1-Cos[t])+Cos[t]}};

In[2]:  rx={{1,0,0},{0,c/Sqrt[b^2+c^2],-b/Sqrt[b^2+c^2]},{0,b/Sqrt[b^2+c^2],c/Sqrt[b^2+c^2]}};
        ry={{Sqrt[b^2+c^2],0,-a},{0,1,0},{a,0,Sqrt[b^2+c^2]}};
        rz={{Cos[t],Sin[t],0},{-Sin[t],Cos[t],0},{0,0,1}};

I wanna to compare if correct equal
Transpose[rx].Transpose[ry].rz.ry.rx
I use Replace and Simplify alternately and the methos is very stupid
and troublesome. Because I have to observe the resulte equation in
advance to decide what "form" or what symbol I should replace it ~
Below is what I do:

In[3]:  aaa = ReplaceRepeated[Simplify[ReplaceAll[Simplify[Transpose[
        rx].Transpose[ry].rz.ry.rx] // MatrixForm, {b^2 -> 1 - a^2 - 
                  c^2, b^4 -> (1 - a^2 - c^2)^2}]], {a^2 + c^2 -> 
                1 - b^2, -a^2 - c^2 -> -1 + b^2}] // MatrixForm

And finally, I found that aaa is the same as the correct matrix, but
it took me a lot of work. Could anyone have good idea or some method
to simplify an equation with some constraints ? Thanks for your help.

           Best   Regards                          Bryan

• Follow-Ups:
  • Re: how to simplify a complicated equation?
    • From: Bob Walker <walkerbg@ieee.org>