Whilst working out some hyperbolic evaluation questions, I've come across this particular one. So far with any question I've come across I've simply tackled it step by step using hyperbolic identities. Before expressing my issue, here's the question.
Given that $\tanh x = \frac{2}{3}$, evaluate $\cosh x$, $\cosh 2x$ and $\tanh 2 x$
After seeing this I simply proceeded by trying to find out which is the correct identity to use to solve this question; however, I can't seem to find, or at least decide, which is the correct identity. My assumption is that I need to go through a series of identities to find out that matches my questions needs. The definition
$$\cosh x := \frac{ e^x + e^{-x}}{2}$$ was the first that came to mind, however I this clearly doesn't fit. The identity $$\tanh x = \frac{\sinh x}{\cosh x}$$ doesn't get me anywhere either. Nor does $$ \tanh^2 x + \text{sech}^2 x = 1 .$$
Any hint on how to tackle this would be of great help.