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I want Mathematica to simplify Cosh[I x] to Cos[x] if I tell Mathematica that x is Real.

I have tried everything, but it ends up giving Cosh[I x] back.

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closed as too localized by Artes, Silvia, Michael E2, m_goldberg, Oleksandr R. Jun 25 '13 at 17:51

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Dotless "i" is useful for a lot of things, but I wouldn't use it for depicting the imaginary unit... – J. M. Jun 13 '12 at 12:47

Simplify (and FullSimplify) gives you the result you want:

Simplify[Cosh[I x]]
(* Cos[x] *)

TrigReduce is another helpful function to remember when working with trigonometric functions.

TrigReduce[expr] rewrites products and powers of trigonometric functions in expr in terms of trigonometric functions with combined arguments.

TrigReduce[Cosh[I x]]
(* Cos[x] *)

You can use TrigExpand to expand out trigonometric functions:

enter image description here

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Thanks, Could you explain how I could expand on this technique to "Expand and simplify coth(x+I y) assuming that x and y are real." – JONASS Jun 13 '12 at 6:28
@JONAS: ComplexExpand[Coth[x + I y]]? – J. M. Jun 13 '12 at 6:32
@JONASS Next time, you should make sure that you don't have such trivial errors before posting here. You can tell the difference between Ix and I x by looking at the color of I — it should be black, assuming you're using the standard scheme and x and Ix aren't symbols known to the kernel. – R. M. Jun 13 '12 at 6:36
Built-in rewrite rules can "simplify" that expression, so e.g Cosh[I x] yields Cos[x] while Cos[I x] returns Cosh[x]. So we don't have to use anything of Simplify or ComplexExpand, etc. – Artes Jun 13 '12 at 8:39
@R.M An anecdote regarding your last comment: We presented a very nice ppt which took us a month to prepare in order to win a very interesting contract. The presentation was colorful and plenty of well designed graphs that depicted our strong points in a convincing way. And we knew we were #1 in the contractors list. After a good show, we lost. A few weeks later, someone from the client told us that the big boss is color blind, and considered our presentation extremely confusing and our lack of research about our audience just inadequate for such a contract. – Dr. belisarius Jun 13 '12 at 12:01

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