$\frac{2x}{\sinh(2\tanh x)}<(\cosh x)^2<\frac{2x}{\sinh(2\tanh x)}+x\sinh (2x)$

Question

I am given a question to prove that

$$\frac{2x}{\sinh(2\tanh x)}<(\cosh x)^2<\frac{2x}{\sinh(2\tanh x)}+x\sinh (2x)$$

I know people will ask what is your attempt/idea. Honestly I have no idea how to solve it. Welcome everyone to give hints/ideas/approach for this question. Thanks in advance.