mathematica, Simplify trig function

if say i have a function given :

               singlepattern = Cosh[theta] + Cosh[3theta]

How do i get a rational expression in terms of x of the function if i want to substitute Cosh[theta] by

               "Cosh[theta] = ( x )/开发者_如何学Go 2 " 

expression?

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I retagged the question as a homework. You should look into ChebyshevT polynomials. It has the property that ChebyshevT[3, Cos[th] ]==Cos[3*th]. So for your problem the answer is

In[236]:= x/2 + ChebyshevT[3, x/2]

Out[236]= -x + x^3/2

Alternatively, you could use TrigExpand:

In[237]:= Cos[th] + Cos[3*th] // TrigExpand

Out[237]= Cos[th] + Cos[th]^3 - 3 Cos[th] Sin[th]^2

In[238]:= % /. Sin[th]^2 -> 1 - Cos[th]^2 // Expand

Out[238]= -2 Cos[th] + 4 Cos[th]^3

In[239]:= % /. Cos[th] -> x/2

Out[239]= -x + x^3/2

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EDIT The reason the above has to do with the explicit question, is that Cosh[theta] == Cos[I*u] for some u. And since u or theta are formal, results will hold true.

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Use Solve to solve for theta, then substitute, Expand, and Simplify:

In[16]:= TrigExpand[Cosh[3 theta] + Cosh[theta]] /. 
  Solve[Cosh[theta] == (x)/2, theta] // FullSimplify

During evaluation of In[16]:= Solve::ifun: Inverse functions are being used by Solve,
so some solutions may not be found; use Reduce for complete solution information. >>

Out[16]= {1/2 x (-2 + x^2), 1/2 x (-2 + x^2)}

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This might interest you:

http://www.wolframalpha.com/input/?i=cosh%28x%29+%2B+cosh%283*x%29