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November 6th, 2017, 01:06 AM  #1 
Senior Member Joined: Jan 2015 From: usa Posts: 101 Thanks: 0  Simplification of a formula
We consider a fixed parameter $\theta>0$. For all $t>0$ we note: $$u(t)=\frac{\sinh\big(\frac{t}{2}\cosh(\theta)\bi g)}{\cosh(\theta)}$$ $$A(t)=\frac{\sqrt{\cosh^2(\theta)u^2(t)+1}1}{\cosh^2(\theta)}\Big(\cosh\Big(\frac{t}{2}\Big)1\Big)$$ $$f(t)=\ln\Big(1\frac{2A(t)}{u(t)+\sinh(t)+A(t)}\Big)$$ For all $\theta>0$ the equation $$2\sinh\Big(cosh(\theta)t\Big)\sinh\Big(t\Big)=1$ $ where $t>0$ has a unique solution that we note $t^*>0$. I want to know if we can simplify $f(t^*)$ or either find a constant $c>0$ such that $f(t^*)\ge c$?? Please help me . Thanks. 

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