Given $\sinh x$, find the exact value of $\cosh x, \cos 2x$ and $\tan 2x$ Given $\sinh x = 8/14$, find the exact value of $\cosh x, \cos 2x$ and $\tan 2x$.
I have been getting two answers which has made me confused. I keep getting $\sqrt{65/7}$ or $\sqrt{4/7}.$
That's what I got:  $$\cosh^2 x - \sinh^2 x = 1$$ $$\cosh^2 x = 1 + \sinh^2 x$$ $$\cosh^2 x = 1 +\left(\frac{8}{14}\right)^2$$ $$\cosh^2 x=\frac{65}{49}$$ $$\cosh (x) = \sqrt{\frac{65}{49}}$$ $$\cosh x = \frac{\sqrt{65}}{7}$$

You are right!  Just use $$\cosh^2 x = 1+\sinh^2 x.$$
Now it only amounts to keeping track of what $\sinh x$ is and carefully doing the calculation.  Is it $\sinh^2 x  = \left(\frac{8}{13}\right)^2$ or is it $\sinh^2 x  = \left(\frac{8}{14}\right)^2$?
UPDATE:  With $\sinh x = \frac{4}{7}$:
$$\cosh^2 x = 1+ \left(\frac{4}{7}\right)^2 = 1+ \frac{16}{49}$$
so
$$\cosh x = \frac{\sqrt{65}}{7}.$$
