For fixed $\theta>0$ we consider the function:

$f(t)=\cosh(\frac{t}{2}\cosh(\theta))\cosh(\frac{t}{2})-\Big(\cosh(\frac{t}{2}\cosh(\theta))+\cosh(\frac{t}{2})\Big)$ for all $t\ge 0$

I want to study the sign of $f$ in $[0,+\infty[$

I calculated the derivative of $f$ and i find :

$f'(t)=\frac{1}{2}\cosh(\theta)sh(\frac{t}{2}\cosh(\theta))\Big(\cosh(\frac{t}{2})-1\Big)+\frac{1}{2}\sinh(\frac{t}{2})\Big(\cosh(\frac{t}{2}\cosh(\theta))-1\Big)\ge 0$ for all $t\ge 0$

Then $f(t)\ge f(0)=-2$

But i woud like to find $t$ for which $f(t)\ge 0$ and $t$ for which $f(t)\le 0$

Please help me. Thanks