I cannot for the life of me work this one out, I am getting very close, but I can't take it further, I must have made a mistake somewhere down the line, anyone mind checking my work? Couldn't get a hold of my prof today

Question: Show that $\displaystyle \frac{sinh 3t}{sinh t}=1+2cosh 2t$.

Here's what I did.

$\displaystyle \frac{sinh(3t)}{sinh(t)}=1+2cosh(2t)$

$\displaystyle \frac{sinh(2t+t)}{sinh(t)}=1+2cosh(2t)$

$\displaystyle \frac{sinh(2t)cosh(t)+cosh(2t)sinh(t)}{sinh(t)}=1+ 2cosh(2t)$

$\displaystyle \frac{2sinh(t)cosh^2(t)+cosh(2t)sinh(t)}{sinh(t)}= 1+2cosh(2t)$

$\displaystyle 2cosh^2(t)+cosh(2t)=1+2cosh(2t)$

I've been trying all sorts of ways to simplify it from here, but none have worked.

Thanks for your time!

Kasper