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By basic trigonometric identities, the expression $\frac{\sinh(x)}{\sqrt{2}\sqrt{1+\cosh(x)}}$ simplifies to $\sinh(\frac{x}{2})$. But Mathematica seemingly don't perform this simplification. I tried restricting the domain of $x$ but that's not the issue, it seems. I also tried to convert it to exponential form and simplifying then bringing back to trigonometric form but even that doesn't work. Removing the square root one gets simplification in terms of half angle. So there seems to be an issue with the square root. The issue remains the same if one uses circular functions instead of hyperbolic. But what is the problem with the square root(if any) and what can I do to make Mathematica to do this simplification automatically?

This is a trivial calculation but I want to automate this with Mathematica because this is part of a much bigger calculation and also for aesthetic/naturalism purposes.

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1 Answer 1

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Clear["Global`*"]
expr = Sinh[x]/(Sqrt[2] Sqrt[1 + Cosh[x]])
(expr /. x -> 2 θ // TrigExpand // 
    Simplify) /. θ -> (x/2) // PowerExpand

Sinh[x/2]

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  • $\begingroup$ Thanks. I will upvote when I have enough reputation :) $\endgroup$
    – Sanjana
    Nov 11, 2023 at 12:46
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    $\begingroup$ Thanks for the accept @Sanjana. I am glad I was able to help you. $\endgroup$
    – Syed
    Nov 11, 2023 at 13:28
  • $\begingroup$ @Syed, can you give some insight into the purpose of the final PowerExpand? Great answer btw; this solved my problem completely! $\endgroup$ Apr 19 at 6:42
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    $\begingroup$ Prior to PowerExpand you get $\sqrt{\cosh ^2\left(\frac{x}{2}\right)} \tanh \left(\frac{x}{2}\right)$. PowerExpand would simplify the squared term inside the square root to allow for the needed cancellation. @CATrevillian $\endgroup$
    – Syed
    Apr 19 at 6:47

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